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    Unsorted algebra problems

    SRC-UNSORTED-ALGEBRA

    Area
    Algebra
    Status
    draft

    830 problems.

    1. Jordan form of \(\begin{pmatrix}1&-1&0\\-1&4&-1\\-4&13&-3\end{pmatrix}\)
    2. Companion matrix of \(x^{2}-x+2\) over \(\mathbb{C}\)
    3. Jordan form of \(\begin{pmatrix}5&-1\\9&-1\end{pmatrix}\oplus\begin{pmatrix}7&-2\\12&-3\end{pmatrix}\)
    4. The nilradical is the intersection of all prime ideals
    5. \(A+\lambda B\) is invertible for all but finitely many \(\lambda\)
    6. Size of a conjugacy class equals the index of the centralizer
    7. \(K(a)=K(a^{2})\) when \(a\) is algebraic of odd degree over \(K\)
    8. Irreducible with a root in a splitting field splits completely
    9. Applications of the class equation
    10. Irreducible polynomials in characteristic \(p\) are \(g(x^{p^k})\) for unique separable \(g\)
    11. Galois group of \(x^2-2\) is \(\mathbb{Z}/2\mathbb{Z}\)
    12. Splitting field and intermediate fields of \(x^{4}-3\) over \(\mathbb{Q}\)
    13. \(a\) is algebraic over \(K(b)\) if \(b\) is algebraic over \(K(a)\) and transcendental over \(K\)
    14. Multiplicative group of a finite field is cyclic
    15. Finite fields are not algebraically closed
    16. Diagonalizability versus invertibility
    17. The stabilizer of a point under a group action is a subgroup
    18. The Galois group of \(x^n-1\) over \(\QQ\) is \((\ZZ/n\ZZ)^\times\)
    19. Finite \(p\)-groups are nilpotent
    20. \(\Inn(G)\) is normal in \(\Aut(G)\)
    21. \(O_p(G)\) is the unique maximal normal \(p\)-subgroup of \(G\)
    22. Nilpotent groups are solvable
    23. \(H\operatorname{char} K\operatorname{char} G\) implies \(H\operatorname{char} G\)
    24. An irreducible in \(\FF_p[x]\) divides \(x^{p^n}-x\) iff its degree divides \(n\)
    25. An operator with irreducible minimal polynomial is semisimple
    26. A normal \(p\)-subgroup is contained in every Sylow \(p\)-subgroup
    27. Burnside’s orbit-counting lemma
    28. A finite group is solvable iff its composition factors have prime order
    29. No simple group of order \(p^2 q^2\) for primes \(p<q\)
    30. An ideal is maximal iff the quotient is a field
    31. \(A_n\) is simple for \(n\geq 5\)
    32. The center of \(S_n\) is trivial for \(n\geq 3\)
    33. A ring is local iff \(x\) or \(1-x\) is a unit for every \(x\)
    34. A linear operator cycling a basis has minimal polynomial \(x^n-1\) and is diagonalizable
    35. A ring in which every non-unit is nilpotent is local
    36. \(A_n\) is simple for \(n\geq 5\)
    37. Every ideal in a Noetherian ring is finitely generated
    38. Prime ideals are primary
    39. An ideal containing a unit is the whole ring
    40. \(|HK|=|H||K|/|H\cap K|\)
    41. Normality of \(\QQ(\sqrt[3]{2})/\QQ\)
    42. Fundamental theorem of Galois theory
    43. Groups of prime order are cyclic
    44. Algebraic extensions are transitive
    45. \(\QQ(2^{1/3})\) and \(\QQ(\zeta_3 2^{1/3})\) are isomorphic but not equal
    46. Every \(p\)-group has a nontrivial center
    47. The union of all maximal ideals is the set of non-units
    48. \(P\cap H\in\syl_p(H)\) for \(P\in\syl_p(G)\) and \(H\trianglelefteq G\)
    49. \(A_n\) is the unique proper nontrivial normal subgroup of \(S_n\) for \(n\geq 5\)
    50. \(-1\) is the unique element of order \(2\) in the quaternion group
    51. Groups of order \(12\) with a normal subgroup of order \(4\) are isomorphic to \(A_4\)
    52. Quotients of dihedral groups are dihedral
    53. Splitting field of \(x^3-2\) is \(\QQ(\sqrt[3]{2},\zeta_3)\)
    54. If \(\spec(R)\subseteq\maxspec(R)\) then \(R\) is a UFD
    55. Groups of order 99
    56. An \(m\)-cycle is odd iff \(m\) is even
    57. Sylow subgroups of a group of order \(240\), and subgroups of order \(15\)
    58. \(C_G(H)\trianglelefteq N_G(H)\)
    59. Diagonalizable versus distinct eigenvalues
    60. \(\GF(p^d)\leq\GF(p^n)\) iff \(d\) divides \(n\)
    61. Cayley-Hamilton theorem
    62. \(\Inn(G)\cong G/Z(G)\)
    63. Every nonzero ring with identity has a maximal ideal
    64. The minimal polynomial over \(L\) divides the minimal polynomial over \(F\)
    65. \(x^{p^n}-x\) is the product of monic irreducibles in \(\FF_p[x]\) of degree dividing \(n\)
    66. \(A_n\) is generated by \(3\)-cycles
    67. The minimal polynomial divides the characteristic polynomial
    68. \(\ff(R[t])=\ff(R)(t)\)
    69. Algebraicity of \(\alpha\pm\beta\) and \(\alpha\beta^{\pm 1}\)
    70. \(K[\alpha_1,\ldots,\alpha_n]=K(\alpha_1,\ldots,\alpha_n)\) for algebraic \(\alpha_i\)
    71. Galois group of \(x^3+4x+2\) over \(\QQ\) is \(S_3\)
    72. Commuting diagonalizable matrices are simultaneously diagonalizable
    73. Normality is not transitive
    74. The radical of an ideal is the intersection of the primes containing it
    75. Groups of order 10
    76. Generating sets for \(S_n\)
    77. Abelian groups are nilpotent
    78. Every prime ideal is radical
    79. Groups of order \(pq\) with \(q<p\)
    80. Composition factors of finite solvable groups have prime order
    81. The Galois group of \(x^n - 2\) over \(\QQ\)
    82. Splitting fields of separable polynomials are Galois
    83. Intermediate fields of \(\QQ(2^{1/4},\zeta_8)\)
    84. \([A_4,A_4]\cong\ZZ_2^2\)
    85. \(\unitsof{R}\) need not be closed under addition
    86. Every permutation is a product of disjoint cycles
    87. \(H\) characteristic in \(K\trianglelefteq G\) implies \(H\trianglelefteq G\)
    88. A linear operator is diagonalizable iff \(V\) is the direct sum of its eigenspaces
    89. Every proper ideal is contained in a maximal ideal
    90. Cyclic groups are solvable
    91. The four groups of order 28
    92. The radical of an ideal is the intersection of the primes containing it
    93. Groups of order \(pqr\) have a normal Sylow subgroup
    94. If \(H\le N_G(K)\) then \(HK\) is a subgroup
    95. A permutation is odd iff it has an odd number of even cycles
    96. Groups of order \(p^3\) have a normal subgroup of order \(p^2\)
    97. Kernel of conjugation \(G\to\Aut(G)\) is \(Z(G)\)
    98. \(Z(S_n)=1\) for \(n\geq 4\)
    99. Intermediate fields of \(\QQ(\sqrt{2},\sqrt{3})=\QQ(\sqrt{2}+\sqrt{3})\)
    100. \(p\)-groups are nilpotent
    101. \(\nilrad{R}=\rad(0)\)
    102. \(Z(G)=\bigcap_{a\in G}C_G(a)\)
    103. Transitive subgroups of \(S_3\) are \(S_3\) and \(A_3\)
    104. Intersections, products, and sums of ideals are ideals
    105. Prime ideals are irreducible
    106. \(p\)-groups are solvable
    107. The cokernel of \(A\in M_n(\ZZ)\) is finite of order \(|\det A|\) iff \(\det A\neq 0\)
    108. Disjoint cycles commute
    109. The union of two ideals need not be an ideal
    110. \(HK\) is a subgroup of \(G\) iff \(HK=KH\)
    111. Finitely generated modules over a Noetherian local ring are flat iff free
    112. Nontrivial normal subgroups of a finite \(p\)-group meet the center
    113. The number of conjugates of \(H\) equals \([G:N_G(H)]\)
    114. \(x^{p^n}-x\) is the product of monic irreducibles in \(\FF_p[x]\) of degree dividing \(n\)
    115. \(I\) is a prime ideal iff \(R/I\) is an integral domain
    116. The five groups of order 12
    117. \(C_H(x)=H\cap C_G(x)\) for \(H\leq G\)
    118. Non-units in a local ring form a proper ideal contained in the Jacobson radical
    119. \(\QQ\) is not finitely generated as a group
    120. \(\maxspec(R)\subseteq\spec(R)\), with strict containment possible
    121. If \(A\) is finite type over Noetherian \(R\) and finite over \(B\), then \(B\) is finite type over \(R\)
    122. Nontrivial normal subgroups of a finite \(p\)-group meet the center
    123. If \(E/k\) and \(F/k\) are Galois with \(E\cap F=k\), then \(EF/k\) is Galois with group \(G(E/k)\times G(F/k)\)
    124. Stabilizers of group actions are subgroups
    125. A group of order \(p^2 q^2\) has a normal Sylow subgroup
    126. \([\QQ(\sqrt{2}+\sqrt{3}):\QQ]=4\) and \(\QQ(\sqrt{2}+\sqrt{3})=\QQ(\sqrt{2}-\sqrt{3})=\QQ(\sqrt{2},\sqrt{3})\)
    127. \(S_4\) has two normal subgroups \(A_4\) and \(\ZZ_2^2\)
    128. Localization at a prime ideal is a local ring
    129. The stabilizer of an element need not be a normal subgroup
    130. Maximal subgroups of a \(p\)-group are normal
    131. \(C_G(H)\subseteq N_G(H)\leq G\)
    132. \(\mathrm{rad}(IJ)=\mathrm{rad}(I)\cap\mathrm{rad}(J)\)
    133. In a PID, a nonzero \(x\) is irreducible iff \((x)\) is maximal
    134. \([S_n,S_n]=A_n\)
    135. Groups of order \(p^2q^2\) are abelian when \(q\nmid p^2-1\) and \(p\nmid q^2-1\)
    136. Galois group of \(x^{n}-1\) over \(\mathbb{Q}\) is abelian
    137. Fixed-point-free elements, quotients of \(\ZZ^3\) of order \(p^2\), groups of order \(pq\), and cyclic Sylow \(2\)-subgroups
    138. Fixed-point-free elements, quotients of \(\mathbb{Z}^3\), groups of order \(pq\), and cyclic Sylow \(2\)-subgroups
    139. Normal subgroups are unions of conjugacy classes
    140. Vandermonde determinant \(\det A=\prod_{i<j}(\lambda_j-\lambda_i)\)
    141. Cyclic groups, Euler’s totient, and \(\operatorname{Aut}(\mathbb{Z}/n\mathbb{Z})\)
    142. Recognizing direct products
    143. Element orders divide the order of a finite group
    144. Nontrivial normal subgroups of a \(p\)-group meet the center
    145. Transitivity of algebraic, normal, and separable extensions
    146. An irreducible generates any principal ideal containing it in a PID
    147. \(\det(A^{a})=\det(A)^{n-1}\) and \((A^{a})^{a}=\det(A)^{n-2}A\)
    148. \(x\in J(R)\iff 1-xR\subseteq R^\times\)
    149. Rank-nullity theorem via Jordan canonical form
    150. Cayley-Hamilton via invariant flags of an upper-triangular basis
    151. The conjugates of a proper subgroup of a finite group do not cover the group
    152. \(A\) is a field iff \(A\) is a simple ring iff every homomorphism from \(A\) to a nonzero field is injective
    153. Cayley-Hamilton via Jordan canonical form
    154. \([F:K]=6\) and \(\operatorname{Aut}_{K}F\cong S_{3}\) implies \(F\) is a cubic splitting field
    155. Quotient by nilradical is reduced
    156. Whether \(\mathbb{Q}(\sqrt{3+\sqrt{2}})\) is a splitting field over \(\mathbb{Q}\)
    157. Normal Sylow \(p\)-subgroup when \(|G|=p^{e}v\) with \(p>v\)
    158. \(R_p\) is a local ring for \(p\in\operatorname{Spec} R\)
    159. Prime ideals, factorization, and chain conditions in PIDs
    160. Complex normal matrices are unitarily diagonalizable
    161. Trivially intersecting normal subgroups commute
    162. \(x^\ell-1\) divides \(x^m-1\) iff \(\ell\) divides \(m\)
    163. A Dedekind domain is a PID if and only if it is a UFD
    164. Poincaré’s theorem on finite-index subgroups
    165. Finite extension with infinitely many intermediate fields
    166. Intersections, unions, and products of subgroups
    167. Algebraic, normal, and separable extensions over intermediate fields
    168. Irreducible polynomial of every degree over \(\mathbb{F}_{p}\)
    169. Lagrange’s theorem: \(|G|/|H|=[G:H]\) for finite \(G\)
    170. Nonzero nilpotent matrices are not diagonalizable
    171. Every non-unit of \(R\) is contained in a maximal ideal
    172. \(G/Z(G)\) cyclic implies \(G\) abelian; \(G/N\) abelian iff \([G,G]\le N\)
    173. \(x^{\ell}-1\) divides \(x^{m}-1\) if and only if \(\ell\) divides \(m\)
    174. Splitting field of \((x^{2}-2)(x^{2}-3)(x^{2}-5)\) over \(\mathbb{Q}\)
    175. Normal subgroups of \(p\)-groups intersect the center
    176. Nilpotents, units, and zero-divisors in \(R[x]\)
    177. An irreducible \(f\in\mathbb{F}_p[x]\) of degree \(d\) divides \(x^{p^n}-x\) if and only if \(d\) divides \(n\)
    178. Degree of a splitting field of a degree-\(n\) polynomial divides \(n!\)
    179. A ring is local if \(R\setminus\mfm\) consists of units, or if \(1+\mfm\) consists of units
    180. Nontrivial solutions of \(Ax=0\) versus \(\det A=0\) over an integral domain
    181. Recognition theorem for internal direct products
    182. Galois group and intermediate fields of \(\QQ(\sqrt{2+\sqrt{2}})/\QQ\)
    183. Finite fields are perfect; example of an imperfect field
    184. Common eigenvectors and simultaneous diagonalization of commuting endomorphisms
    185. Join of intermediate fields and \(\mathrm{Aut}\) of a Galois extension
    186. Cayley-Hamilton theorem, cokernels of integer matrices, and diagonalizability
    187. \(\sum(-1)^{i}\dim V_{i}=0\) for a finite exact sequence of vector spaces
    188. \(K\)-automorphisms permute the roots of polynomials in \(K[x]\)
    189. Radical, prime, and maximal ideals correspond under \(R\to R/I\)
    190. Size of a conjugacy class divides the order of the group
    191. Galois group of \(x^p-2\) over \(\QQ\)
    192. Splitting field of \(x^{3}-x+1\) over \(\mathbb{F}_{3}\)
    193. \(R[x]\) is a PID if and only if \(R\) is a field
    194. Primitive odd \(n\)th root of unity implies a primitive \(2n\)th root of unity
    195. \(\mathbb{Q}(x)\) is Galois over \(\mathbb{Q}(x^{2})\) but not over \(\mathbb{Q}(x^{3})\)
    196. \(\mathbb{F}_{p^n}\) is the splitting field of \(x^{p^n}-x\) over \(\mathbb{F}_p\)
    197. A ring in which every element is a unit or nilpotent is local
    198. The union of conjugates of a proper subgroup is a proper subset
    199. \([\mathbb{Q}(\zeta_{n}+\zeta_{n}^{-1}):\mathbb{Q}]=\varphi(n)/2\)
    200. A splitting field of separable elements is Galois
    201. The normal closure of a simple extension is its minimal-polynomial splitting field
    202. A finite extension is algebraic and contained in a finite splitting field
    203. The sum of (skew-)symmetric matrices is (skew-)symmetric
    204. \(A/IA\) is an \(R/I\)-module
    205. Unique proper nontrivial normal subgroup of \(S_n\) (\(n\geq 5\)) is \(A_n\)
    206. \((2,x)\) in \(\ZZ[x]\) is not a direct sum of cyclic modules
    207. Translation-invariant subspaces of \(L^1\) as convolution algebras relative to \(L^2\)
    208. Groups of order 56 and actions of \(\ZZ/7\ZZ\) on groups of order 8
    209. The quaternion group has a unique element of order \(2\), namely \(-1\)
    210. Intermediate fields of a finite cyclic Galois extension
    211. Sylow subgroups for distinct primes intersect trivially
    212. Groups of order \(240\): Sylow numbers and subgroups of order \(15\)
    213. Every non-unit lies in a maximal ideal
    214. Maximal subgroups of \(p\)-groups are normal
    215. A non-semisimple \(\CC\)-algebra
    216. Eigenvalues of skew-symmetric matrices
    217. Groups of order \(p^2\) are abelian
    218. \(\Inn(G)\cong G/Z(G)\)
    219. Schur’s lemma
    220. \(\ZZ/m\ZZ\) as a \(\ZZ/(mk)\ZZ\)-module and its dual
    221. Order of \(1+p\) in \((\ZZ/p^2\ZZ)^\times\) is \(p\); a nonabelian group of order \(p^3\)
    222. \(P\cap H\in\syl_p(H)\) for \(P\in\syl_p(G)\) and \(H\trianglelefteq G\)
    223. Classification of abelian groups of a given composite order
    224. Groups of order \(pq\) (\(q<p\)) are never simple, and are cyclic if \(q\nmid(p-1)\)
    225. The integers form a PID
    226. Cokernel of a map \(\ZZ^4\to\ZZ^3\)
    227. Finite division rings are fields, and an infinite noncommutative example
    228. Jordan form of the \(n\times n\) all-ones matrix over \(\CC\)
    229. A normal subgroup of order \(4\) does not force a group of order \(12\) to be \(A_4\)
    230. Torsion matrix cokernels over a PID and an \(R\times F\) example
    231. Galois group of \(x^3+2x+1\)
    232. Algebraic, normal, and separable extensions over intermediate fields
    233. The nilradical is contained in the Jacobson radical
    234. \(C_G(G)=Z(G)\)
    235. Whether \(k[x_1,\ldots,x_n]\) is a PID
    236. Normal subgroups of \(A_4\)
    237. Local rings via localization and units; primes are primary and irreducible
    238. Isomorphic subgroups need not have isomorphic quotients
    239. The center of a \(p\)-group is nontrivial
    240. A commutative unital ring is a field iff its only ideals are \(\{0\}\) and \(R\)
    241. Abelian groups of order 36
    242. Isomorphism classes of subgroups of \(\QQ\)
    243. A group of order \(p^2q\) has a nontrivial normal subgroup
    244. \(\QQ(2^{1\over 3})\) and \(\QQ(\zeta_3 2^{1\over 3})\)
    245. Dirichlet’s theorem on primes in arithmetic progressions
    246. Finite-dimensional representations of compact Lie groups
    247. Artin–Wedderburn theorem
    248. \(\mathbb Z/n\mathbb Z\) as a direct sum of fields
    249. Groups of order \(p^3\) are abelian or have \(|Z(G)|=p\)
    250. Characteristic and minimal polynomials of the Frobenius automorphism
    251. Normal closure and Galois theory of \(\QQ(\sqrt[3]{21})\)
    252. Isomorphic but unequal fields
    253. Structure theorem for PID-modules applied to a linear operator
    254. \(C_H(x)=H\cap C_G(x)\)
    255. Groups of order 8
    256. Irreducible modules over a PID are \(R/(p)\) and indecomposable modules are \(R/(p^n)\)
    257. Noetherian rings and \(\bigcap I^n\)
    258. A degree-\(6\) \(S_3\)-extension is a cubic splitting field, and Galois groups of joins and intersections
    259. Groups of order 15
    260. A PID with a unique prime ideal
    261. Groups of order \(p^2q^2\) have a normal Sylow subgroup
    262. Centers of \(\GL_n(\FF_p)\) and \(\SL_n(\FF_p)\) are scalar matrices
    263. Possible Galois groups of an abelian extension of degree \(540\), and quadratic intermediate extensions \(L/E\)
    264. \(H\operatorname{char} K\trianglelefteq G\) implies \(H\trianglelefteq G\)
    265. Toeplitz operator
    266. Whether \(\QQ(2^{1/3})/\QQ\) is normal
    267. Irreducible polynomials of degrees 7 and 14 over \(\FF_p\)
    268. Every field is a simple ring
    269. Galois group of an irreducible quartic with exactly two real roots is \(S_4\) or \(D_4\)
    270. The centralizer of a subgroup is a subgroup
    271. A non-symmetric polynomial whose square is symmetric
    272. A root field with normal closure of Galois group \(S_n\) is not contained in a radical extension
    273. A field is perfect if every irreducible polynomial is separable
    274. Images of the unit circle under linear transformations of \(\mathbb{R}^2\)
    275. Order, centre, and \(\PSL_2(\FF_3)\) as a permutation group
    276. Torsion elements of an \(R\)-module: definition, an infinite cyclic example, and the torsion submodule over a domain
    277. A PID that is not Euclidean
    278. \(H\operatorname{char} K\operatorname{char} G\) implies \(H\operatorname{char} G\)
    279. \(Z(G)\) is characteristic in \(G\)
    280. A ring in which unique factorization fails
    281. Orbit-stabilizer theorem
    282. No simple group of order \(132\)
    283. The splitting field of \(x^3-2\) is \(\QQ(\sqrt[3]{2},\zeta_3)\)
    284. A real \(5\times 5\) matrix with eigenvalues \(0\), \(1+i\), and \(1+2i\): injectivity, characteristic and minimal polynomials, and fixed points
    285. \(Z(A_n)=1\) for \(n\geq 4\)
    286. Groups of order \(45\) are abelian
    287. \(G/Z(G)\) cyclic implies \(G\) abelian
    288. Solvability of polynomials by radicals
    289. \(\GF(p^n)\) is the splitting field of \(x^{p^n}-x\) over \(\FF_p\)
    290. \(\QQ(\sqrt{2}+\sqrt{3})=\QQ(\sqrt{2},\sqrt{3})\) has degree \(4\) over \(\QQ\)
    291. Proper nontrivial normal subgroups of \(S_4\) are \(A_4\) and \(\ZZ_2^2\)
    292. Invariant factors of \(R/(r)\oplus R/(s)\) over a PID
    293. \(S_4\) is solvable and nonabelian
    294. Galois group of \(x^4-3\)
    295. Roots of an integer polynomial with Galois group \(S_n\)
    296. Subgroup lattice of \(S_3\)
    297. A finite group is not the union of conjugates of a proper subgroup; \(S_n\) has a derangement
    298. Galois group of \(x^p-2\) over \(\QQ\) for odd primes \(p\)
    299. \(HK\) is a subgroup iff \(HK=KH\)
    300. \(\mathrm{Gal}(\QQ(\sqrt{2},\sqrt{3})/\QQ)\)
    301. Unique intermediate field of each degree in a cyclic Galois extension
    302. The discriminant of a polynomial is a polynomial in the coefficients, via symmetric polynomials
    303. A Galois group that is neither \(S_n\) nor \(A_n\)
    304. \(p\)-groups are nilpotent
    305. Representations and the group algebra of \(\ZZ\)
    306. Matrices satisfying \(A^2=A\) are diagonalizable
    307. A long exact sequence from a short exact sequence of abelian groups and a third group
    308. Galois extension
    309. A group is solvable iff its derived series terminates
    310. If \(\exp(A)=B\in\SL_n(\RR)\), must \(A\in\SL_n(\RR)\)?
    311. \(A_n\) is simple for \(n\geq 5\)
    312. Cayley’s theorem and embeddings into \(S_m\) with \(m<n\)
    313. Sylow theorems
    314. \(\ZZ_n[m]\cong\ZZ_{(m,n)}\)
    315. The quaternion group and the number of elements of each order
    316. \([\QQ(\zeta+\zeta^{-1}):\QQ]=\phi(n)/2\)
    317. Maximal ideals in a countable product of copies of a commutative ring
    318. \(R/(k)\cong sA\) and \(R/(s)\cong A[s]\) when \(A=Ra\) is cyclic with annihilator \((r)\) and \(r=sk\)
    319. Subgroups of index two are normal
    320. Image \(pA\) and kernel \(A[p]\) of multiplication by \(p\)
    321. Whether algebraic, normal, and separable remain so over \(E\)
    322. \(\mathrm{Gal}(x^3+4x+2/\QQ)\cong S_3\)
    323. Whether \(A_4\) is simple, and conjugacy in \(S_4\) and \(A_4\)
    324. Every simple \(R\)-module is cyclic
    325. Irreducibility of \((x^p-1)/(x-1)\) over \(\QQ\)
    326. Rational roots of monic polynomials in \(\ZZ[x]\) are integers
    327. Whether representations conjugate at each \(g\in G\) are isomorphic
    328. Whether inducing an irreducible representation of a subgroup remains irreducible
    329. Maximal ideals are prime; the converse fails
    330. In a finite ring every element is a unit or a zero-divisor
    331. The nilradical is contained in the Jacobson radical
    332. The nilradical is the intersection of all prime ideals
    333. \(R/\nilrad{R}\) is reduced
    334. Irreducible minimal polynomial implies every invariant subspace has an invariant complement
    335. \(\mathrm{Hom}_{\ZZ}(\ZZ_m,A)\cong A[m]\)
    336. Frobenius of \(\FF_{p^n}/\FF_p\) and its characteristic polynomial
    337. \(\CC^2\) as a cyclic torsion \(\CC[x]\)-module via \(\begin{pmatrix}1&2\\2&1\end{pmatrix}\)
    338. Groups of order \(p^2q\)
    339. Rational canonical form
    340. \(H\le N_G(K)\) is sufficient but not necessary for \(HK\) to be a subgroup
    341. \(\mathrm{Hom}_R(R,R)\cong R^{\mathrm{op}}\)
    342. Definition of a splitting field, and a finite extension of \(\FF_q\) as a Galois splitting field
    343. \(\mathrm{Gal}(M/L)\) is normal in \(\mathrm{Gal}(M/K)\)
    344. Definition of a splitting field
    345. Non-isomorphic groups can have isomorphic automorphism groups
    346. Groups of order \(p^3\)
    347. Groups of order \(p^2\) and of order \(pq\)
    348. \(A_n\) is normal in \(S_n\)
    349. \(\QQ\) is not a finitely generated group
    350. A subgroup of index equal to the smallest prime dividing \(|G|\) is normal
    351. Euler totient of primes and composites
    352. Every permutation in \(S_n\) is a product of disjoint cycles
    353. \(\phi(n)\) is even for \(n>2\)
    354. \(S_p\) is generated by a \(p\)-cycle and a transposition
    355. Nilradical is intersection of primes
    356. Burnside’s lemma
    357. Diagonalizable does not imply distinct eigenvalues
    358. \([K(\alpha):K]\) equals the degree of the minimal polynomial of algebraic \(\alpha\)
    359. Order of a cyclic module over a commutative ring
    360. Definitions of UFD and PID, and a UFD that is not a PID
    361. Representations of \(S_3\) and restriction to \(S_2\)
    362. Galois groups of irreducible cubics
    363. A finite group is solvable iff its composition factors have prime order
    364. Representations and character table of \(D_4\)
    365. Conjugacy classes in \(S_n\)
    366. \(\operatorname{Frac}(R[t]) = \operatorname{Frac}(R)(t)\)
    367. Galois group of \(x^{15}+2\) over \(\QQ\) is a semidirect product of a Sylow \(2\)-subgroup with \(\ZZ/15\ZZ\)
    368. \(R/(p)\) is a field when \((p)\) is prime in a PID
    369. Disjoint cycles commute
    370. The integers \(n\) with \(\phi(n)=2\) are \(3,4,6\)
    371. Diagonalizable matrices have squarefree minimal polynomials
    372. Maximal implies prime
    373. \(\ZZ^4/N\cong\ZZ\oplus\ZZ/(3)\oplus\ZZ/(6)\oplus\ZZ/(6)\) by Smith normal form
    374. Equivalence of \(A\mathbf{x}=\mathbf{b}\) with the corresponding system of linear equations
    375. Abelian groups of order 200
    376. An intermediate field of a Galois extension is normal iff its Galois group is a normal subgroup
    377. Finite fields, subfields, extensions, and automorphisms
    378. Dyadic rationals form a proper noncyclic subgroup of \((\QQ,+)\)
    379. If \(M/K\) is Galois, whether \(L/K\) or \(M/L\) is Galois
    380. Fundamental theorem of Galois theory
    381. An endomorphism of \(\RR^5\) with eigenvalue \(0\) is neither injective nor surjective
    382. Galois theory of \(x^{2n}-2\)
    383. Groups of order \(99\)
    384. A diagonalizable matrix has a cyclic vector if and only if its eigenvalues are distinct
    385. Diagonalizable iff the space is a direct sum of eigenspaces
    386. Hilbert’s theorem 90
    387. \(NH\) is a subgroup when \(N\) is normal
    388. Cycle type, order, and sign of \((4\,2\,1)(6\,1\,3\,2)\)
    389. Degree of a field extension
    390. Matrices with the same Jordan form are similar
    391. Galois theory of cyclotomic extensions
    392. Galois group of \(x^3-3x-3\) over \(\QQ\) is \(S_3\)
    393. Stabilizers for \(\SL_2(\RR)\) acting on \(\RR^2\) and by Möbius transformations
    394. Stabilizers of points in the same orbit are conjugate
    395. Fundamental theorem of symmetric polynomials
    396. The structure theorem for modules over a PID and conjugacy classes in \(\mathrm{GL}(V)\)
    397. \(F(\alpha)/F\) is finite iff \(\alpha\) is algebraic over \(F\)
    398. Groups of order 35
    399. Unique subfield of order \(p^d\) in a field with \(p^n\) elements
    400. Galois group of \(x^4-2x^2+9\)
    401. \(|HK| = |H||K|/|H \cap K|\)
    402. Groups of order 12, and two nonisomorphic subgroups of the same order
    403. Rational and Jordan forms with given minimal polynomials over \(\mathbb{Q}\) and \(\mathbb{C}\)
    404. \(p\)-adic numbers and valuations
    405. Polar decomposition of operators on a Hilbert space
    406. Factorisation in \(R[x_1,\ldots,x_n]\)
    407. Quotients by maximal ideals are fields and quotients by prime ideals are integral domains
    408. A normal \(p\)-subgroup is contained in every Sylow \(p\)-subgroup
    409. The field with 25 elements
    410. The smallest extension of \(\FF_q\) splitting every polynomial of degree at most \(4\)
    411. \(A_n\) is generated by \(3\)-cycles
    412. Classification of finitely generated modules over \(\mathbb{Z}\), PIDs, and Dedekind rings
    413. Prime and maximal ideals of \(\ZZ[x]\)
    414. Galois group of \(x^2-2\)
    415. Finite type over a Noetherian ring through a finite algebra
    416. Lower and upper central series, nilpotent groups, and solvable groups
    417. A square matrix is similar to its transpose
    418. Representations of finite groups: irreducibles, complete reducibility, and invariant inner products
    419. Group representations and their traces
    420. Chinese remainder theorem
    421. Principal and prime ideals in \(\CC[x,y]\)
    422. Galois group of \(x^3-2\) over \(\QQ\)
    423. Structure theorem for finitely generated modules over a PID
    424. Every \(p\)-group is nilpotent
    425. \(\Hom_{\ZZ}(\ZZ/2\ZZ,-)\) is not right exact
    426. Derived subgroup equals the centre for nonabelian groups of order \(p^3\)
    427. Cayley–Hamilton theorem
    428. Semisimple algebras and the Artin–Wedderburn theorem
    429. \(K(u)/K\) is not Galois and \(\Aut(K(u)/K)\) is trivial for a single root of an irreducible of degree \(n\)
    430. Discriminant of a polynomial
    431. Number of irreducible polynomials of degree \(6\) over \(\FF_2\)
    432. A \(p\)-group has nontrivial center
    433. A field homomorphism is zero or injective
    434. A linear operator with \(1\notin\spec(L)\) has unique fixed point \(0\)
    435. The torsion subset of a module over an integral domain is a submodule
    436. Jordan and rational canonical forms for \(\chi_A=(x-1)^2(x+1)^2\) and \(p_A=(x-1)(x+1)^2\)
    437. Minimal and characteristic polynomials of a \(5\times 5\) real matrix with eigenvalues \(0,1\pm i,1\pm 2i\)
    438. Eigenvalues of a symmetric matrix
    439. If \(K/F\) is cyclic and \(E/F\) is normal then \(E/F\) and \(K/E\) are cyclic
    440. Simple abelian groups are cyclic
    441. \(C_G(H) \subseteq N_G(H) \leq G\)
    442. \(\Endo_R(A)\) is a ring
    443. Galois group and intermediate fields of \(x^3-7\) over \(\QQ\), and Galois groups over \(\RR\) and \(\FF_{13}\)
    444. Colorings of a tetrahedron with \(C\) colors up to symmetry
    445. For symmetric \(A,B\), the product \(AB\) is symmetric iff \(AB=BA\); \(BB^t\) and \(B+B^t\) are symmetric
    446. Spectral theorem for real symmetric matrices
    447. Jordan–Hölder theorem
    448. Groups of order \(p^n\) are solvable, and groups of order \(p^r q^s\)
    449. The rings and fields \(K[x]\), \(K(x)\), \(K[\alpha]\), and \(K(\alpha)\)
    450. The splitting field of a separable polynomial is Galois
    451. Rational canonical forms of \(3\times 3\) matrices over \(\QQ\) annihilated by \((x^2+2)(x-1)^3\)
    452. Linear actions of \(\ZZ/5\ZZ\), \(S_3\), and dihedral groups on complex vector spaces
    453. Groups of order 14
    454. Ring of integers and integrality over \(\ZZ\)
    455. Induced representations
    456. When \(A^n\to 0\)
    457. Inner automorphisms preserve conjugacy of subgroups
    458. Cauchy’s theorem
    459. \(A_n\) is simple for \(n \geq 5\)
    460. Similarity and equivalence of matrices are equivalence relations
    461. \(\operatorname{Hom}_{\ZZ}(\ZZ/m\ZZ, \ZZ) = 0\)
    462. Element orders in the quaternion group, and quadratic subfields of a \(Q_8\)-extension of \(\QQ\)
    463. A polynomial of degree \(2\) or \(3\) in \(k[x]\) is irreducible iff it has no root in \(k\)
    464. Witt’s theorem on real quadratic forms
    465. Rational canonical forms from specified minimal polynomials in dimensions \(6\) and \(10\)
    466. Euclidean domains and PIDs
    467. Composition series, solvability, and nilpotence
    468. \(\ZZ/p\ZZ\otimes\ZZ/q\ZZ\) for distinct primes \(p\) and \(q\)
    469. Constructible numbers
    470. Nonisomorphic groups with equivalent representation theory
    471. Smith normal form of quotients of a free abelian group, and the Euclidean-domain case
    472. Eigenvalues of Hermitian matrices are real, and eigenvalues of unitary matrices have modulus \(1\)
    473. Quotients of quadratic integer rings by nonzero prime ideals
    474. Algebraic and transcendental elements
    475. The transitive subgroups of \(S_3\) are \(S_3\) and \(A_3\)
    476. Tensor product
    477. Abelian groups of order 16
    478. Consider a nonconstant function between two compact Riemann Surfaces.
    479. An \(m\)-cycle is odd iff \(m\) is even
    480. Evaluation makes \(A\) a module over \(\mathrm{End}_R(A)\)
    481. A subgroup that meets every conjugacy class is the whole group
    482. The \(p\)-torsion \(A[p]\) is an \(R/(p)\)-module
    483. \(SL_2(\ZZ)\) is not solvable
    484. Diagonalizable matrices are dense over \(\CC\) and Zariski-dense over an algebraically closed field
    485. Groups of order \(28\)
    486. Eigenvalues of a linear map preserving a nondegenerate alternating form come in pairs \(k,1/k\)
    487. Necessary and sufficient conditions for the Galois group of a degree \(5\) polynomial to have order divisible by \(3\)
    488. No simple group of order \(p^2 q^2\)
    489. Jordan form of the \(n\times n\) all-ones matrix
    490. Minimal polynomial divides the characteristic polynomial
    491. \(\mathrm{Gal}(\QQ(\sqrt{n_1},\ldots,\sqrt{n_m})/\QQ(\sqrt{n_1}+\cdots+\sqrt{n_m}))\)
    492. Nilpotent matrix
    493. Factor 6 in two different ways in \(\ZZ[\sqrt{-5}]\)
    494. A Galois-theoretic proof of the fundamental theorem of algebra
    495. The center of \(S_n\) is trivial for \(n \geq 3\)
    496. Definition of a splitting field; a finite extension of a finite field as a splitting field, and that the extension is Galois
    497. Nontrivial centre of a group of order \(p^3\)
    498. Injective module
    499. \(Z(G) = \bigcap_{a \in G} C_G(a)\)
    500. Number of irreducible representations of \(S_n\)
    501. \([\QQ(2^{3/2}):\QQ]\)
    502. The kernel of a homomorphism is a normal subgroup
    503. The class equation
    504. A group of order \(231\) has a normal Sylow \(7\)-subgroup and a central Sylow \(11\)-subgroup
    505. Existence and uniqueness of \(AX=b\) by rank
    506. Finite field extensions are algebraic
    507. Automorphisms of \(\CC\)
    508. The alternating group \(A_n\) is generated by \(3\)-cycles
    509. Impossibility of trisecting an angle, doubling a cube, and squaring a circle
    510. Whether a formal power series ring is a UFD
    511. Dedekind domains and class numbers
    512. Centralizer of a Jordan block
    513. Lie groups, Peter–Weyl, and representations of \(\ZZ\)
    514. Solvable groups, a nonabelian example, solvability of \(A_4\), and normality of its index-\(3\) subgroup
    515. Infinite simple groups have no finite-index subgroups
    516. Pairs \((n,p)\) for which \(\SL_n(\FF_p)\) is solvable
    517. An example of a flat module
    518. Projective module
    519. The Galois group of \(x^n - 2\) over \(\QQ\)
    520. Centre of \(S_n\)
    521. A subgroup of a finite \(p\)-group generating \(H/[H,H]\) equals \(H\)
    522. Normality of subgroups is not transitive
    523. Image, kernel, cosets, and the first isomorphism theorem
    524. Whether \(R\) a UFD implies \(R[x]\) a UFD
    525. Geometric diagonalisation of a quadratic form
    526. Naturality of the bidual evaluation map
    527. Every \(p\)-group is solvable
    528. If \(A=Ra\) with \(ra=0\) and \((r,s)=(1)\), then \(A=sA\) and \(A[s]=0\)
    529. A polynomial with Galois group \(S_3\)
    530. Subgroups of \(\ZZ^2\)
    531. If \(Q/N\) is a Sylow \(p\)-subgroup of \(G/N\) then \(Q\) contains a Sylow \(p\)-subgroup of \(G\), and every Sylow \(p\)-subgroup of \(G/N\) is such an image
    532. Groups of order \(p^3\) have a normal subgroup of order \(p^2\)
    533. Matrices with \(\tr(A^k)=0\) for all \(k\)
    534. A radical extension is radical over any intermediate field
    535. Stabilizers are subgroups
    536. Hom–tensor adjunction
    537. Why the character table is square
    538. Proper subfields of \(\CC\) isomorphic to \(\CC\)
    539. The Galois group of \(x^4-2\) over \(\QQ\)
    540. \(S_3\) is not nilpotent
    541. Matrices satisfying a given polynomial over algebraically closed and finite fields
    542. Square roots of matrices over \(\CC\)
    543. Generating sets for \(S_n\)
    544. \(L/F\) is algebraic if \(K/F\) and \(L/K\) are algebraic
    545. Mod by nilradical to kill nilpotents
    546. If every irreducible in \(F[x]\) is separable then every element of \(F\) is a \(p\)-th power
    547. Finitely generated flat modules over Noetherian local rings are free
    548. A \(p\)-cycle and a transposition generate \(S_p\) for \(p\) prime
    549. Operators to which the spectral theorem for symmetric matrices generalizes
    550. \(C_G(H)\) is normal in \(N_G(H)\)
    551. \(Z(R)\cong Z(M_n(R))\) via \(r\mapsto rI_n\)
    552. Conjugacy of matrices over a field and over its algebraic closure
    553. Orbits, stabilizers, and kernels of the standard \(G\)-actions
    554. The character table of \(S_4\)
    555. \(\Aut(S_3)=\Inn(S_3)\cong S_3\)
    556. Commutator subgroup of \(S_n\) is \(A_n\)
    557. Character table
    558. Irreducible representations of finite abelian groups
    559. Eisenstein’s criterion
    560. Jacobson radical
    561. Groups of order \(pq\) need not be nilpotent
    562. Conjugation on Sylow \(p\)-subgroups; \(|G|\) divides \(n_p!\) if \(G\) is simple; no simple group of order \(ap^k\) for \(1\le a\le 4\)
    563. \(\hom_R(A, B)\) under pointwise addition is an abelian group
    564. \(x^{p^n}-x\) is the product of all monic irreducibles of degree dividing \(n\)
    565. Nilpotent groups have nontrivial centers
    566. The center of \(M_n(R)\) is \(Z(R)I_n\)
    567. Infinitely many primes congruent to \(1\bmod m\) via cyclotomic polynomials
    568. Groups of order 55
    569. Number of conjugates of a subgroup equals \([G:N_G(H)]\)
    570. Conjugacy of stabilizers along an orbit
    571. Groups of prime order are cyclic and simple
    572. \(HK\) is a subgroup when \(H\le N_G(K)\); a counterexample when \(K\) is not a subgroup
    573. Groups of order 21
    574. Distinct Sylow subgroups intersect trivially; normal subgroups of \(p\)-groups meet the center
    575. Symmetry group of the icosahedron
    576. Polynomial rings over integral domains are integral domains
    577. \(g^{\lvert G\rvert}=e\) for every \(g\in G\)
    578. The four lemma
    579. An ideal of a PID containing an irreducible \(a\) equals \(Ra\)
    580. Sufficient conditions for a degree \(5\) polynomial to have Galois group \(S_5\) over \(\QQ\)
    581. An irreducible cubic with Galois group not in \(A_3\) has Galois group \(S_3\)
    582. Elementary matrices preserve solutions of \(Ax=b\)
    583. Given a skew-symmetric/skew-Hermitian matrix S, show that
    584. \((\FF_9)^\times\)
    585. \(Ax=0\) has a nontrivial solution iff \(\rank(A)<m\)
    586. A permutation is odd iff it has an odd number of even-length cycles
    587. Groups of order 14
    588. Left, right, and two-sided ideals in \(M_n(K)\), and simple ideals in \(\Endoo_K(V)\)
    589. The Galois group of an irreducible quintic with two non-real roots is \(S_5\)
    590. Finite integral domains are fields
    591. Definition and an example of a Noetherian ring
    592. Recognizing direct products
    593. Separable extensions
    594. Transitive subgroups of \(S_4\) are \(S_4\), \(A_4\), \(D_4\), \(\ZZ_2^2\), and \(\ZZ_4\)
    595. Intermediate fields of a Galois extension with group \(\ZZ_{42}\)
    596. \(\ZZ\) is initial among unital rings, so every abelian group has a unique \(\ZZ\)-module structure
    597. Galois group of \(x^5-2\)
    598. Groups of order four as Galois groups over \(\QQ\)
    599. Groups of order \(pq\) with \(p>q\) have a proper nontrivial normal subgroup
    600. \(\mathrm{Ext}\) of cyclic groups
    601. \(\operatorname{rank}(F_1\oplus F_2)=\operatorname{rank} F_1+\operatorname{rank} F_2\) for free modules with the invariant dimension property
    602. A group of order \(p^2q\) has a normal Sylow subgroup
    603. A field extension with finitely many intermediate fields is simple
    604. Orders of \(\GL_n(\FF_p)\) and \(\SL_n(\FF_p)\)
    605. The number of subgroups conjugate to \(H\le G\) divides \(|G|\)
    606. Normal subgroups of \(p\)-groups meet the center nontrivially
    607. Integral closure of \(\ZZ\) in \(\QQ(i)\)
    608. Finite groups as Galois groups
    609. An algebraic extension \(C/F\) splits every polynomial over \(F\) if and only if every algebraic extension of \(F\) embeds in \(C\)
    610. Separable polynomials of prescribed degree
    611. No group of order 90 is simple
    612. Subgroups of order 5 in \(S_5\) are transitive
    613. Number of invertible matrices over \(\ZZ/p\ZZ\)
    614. Group rings
    615. Intermediate fields of \(\QQ(2^{1/4},\zeta_8)\)
    616. Units or Zero Divisors
    617. Characteristic subgroups are normal
    618. Free modules over an integral domain are torsion-free; \(\QQ\) is torsion-free but not free as a \(\ZZ\)-module
    619. There is no simple group of order 148
    620. Galois group of \(x^n-1\) over \(\QQ\) as a function of \(n\)
    621. The Hilbert basis theorem
    622. Symmetric polynomials
    623. \(Z(G)\subseteq C_G(H)\subseteq N_G(H)\)
    624. \(N/C\) theorem
    625. A subgroup of index \(2\) is normal; the same for index the smallest prime dividing \(|G|\)
    626. Jordan form of an operator on a \(6\)-dimensional space with \(T^6=0\) and \(T^5\neq 0\), and similarity of any two such operators
    627. \(\alpha\pm\beta\) and \(\alpha\beta^{\pm 1}\) are algebraic over \(F\) whenever \(\alpha\) and \(\beta\) are
    628. Subgroups of \(\ZZ^3\) and the structure of the quotient
    629. Every finitely generated torsion-free abelian group is free abelian
    630. Representations of \(G\) lifted from \(G/H\) have common kernel \(H\), and the commutator subgroup
    631. Prime elements of \(\ZZ[i]\)
    632. \(F[x]\oplus F\) as the cokernel of a \(2\times 2\) Smith normal form matrix
    633. Prime ideals in \(k[x,y]\) and the Nullstellensatz
    634. Solving \(\exp(A)=B\) over \(\CC\) and over \(\RR\)
    635. Sylow \(p\)-subgroups of \(\GL_3(\FF_p)\), their conjugates, and their normalizers
    636. No group of order 36 is simple
    637. Ruler-and-compass constructibility of a regular \(n\)-gon
    638. \((\RR,+)\cong(\RR_{>0},\times)\), but not for \(\QQ\)
    639. The cokernel of \(A\in M_n(\mathbb{Z})\) is finite iff \(\det A\neq 0\), with order \(|\det A|\)
    640. Examples of local rings
    641. A cubic with Galois group \(S_3\)
    642. Wedderburn’s theorem and rational group algebras
    643. Quotients of dihedral groups are cyclic or dihedral
    644. Hilbert’s Nullstellensatz
    645. Solvability by radicals, \(S_5\), and simplicity of \(A_5\)
    646. Normality of field extensions is not transitive
    647. \(O_p(G)\) is the maximal normal \(p\)-subgroup of \(G\)
    648. \(D_4\)-invariant polynomials in two real variables
    649. \(\Inn(G)\) is normal in \(\Aut(G)\)
    650. Characters of the multiplicative group of a finite field
    651. Gauss’s lemma: factorisation in \(\QQ[x]\) versus \(\ZZ[x]\)
    652. Kernels of irreducible representations and exactness of induction
    653. Solvable groups have a nontrivial normal subgroup
    654. The number of irreducible polynomials of degree \(4\) over \(\FF_2\)
    655. Self-adjoint operators have real eigenvalues and are unitarily diagonalizable, and Schur’s theorem
    656. Galois group of \(x^7-1\) over \(\QQ\)
    657. Bilinear forms and orthogonal matrices
    658. Subgroups of \(F_2\)
    659. Every element of a finite-dimensional \(k\)-algebra is integral over \(k\), and a non-zero-divisor is a unit
    660. A Lie group with no faithful finite-dimensional representation
    661. Conjugacy classes in \(S_5\) by cycle type, corresponding to partitions of \(5\)
    662. If \(N\) and \(G/N\) are solvable then \(G\) is solvable
    663. Separable extensions are normal iff they are splitting fields
    664. Composition factors of finite solvable groups have prime order
    665. The kernel of the conjugation homomorphism \(G\to\mathrm{Aut}(G)\) is \(Z(G)\)
    666. Galois theory exercises: splitting fields, intermediate extensions, finite fields
    667. Irreducible varieties
    668. Structure theorem for finitely generated modules over a PID
    669. Cyclotomic specialization of \(\ZZ[t]/(t^p-1)\)
    670. Cosets of a subgroup all have the same cardinality
    671. Normal subgroups
    672. Frobenius reciprocity
    673. Fundamental theorem of cosets: \(xH=yH\iff x^{-1}y\in H\iff y^{-1}x\in H\)
    674. Examples of simple groups
    675. Perfect fields
    676. Galois group of \(x^2+1\) and the Gaussian integers
    677. Orders of elements of \(\mathrm{SL}_2(\ZZ)\)
    678. Number of elements of order \(4\) in \(\Aut(\ZZ/20)\)
    679. The order of an element divides the order of the group
    680. When conjugacy classes of \(S_n\) split in \(A_n\)
    681. The centralizer \(C_G(s)\) is a subgroup in which \(\langle s\rangle\) is normal
    682. Groups of order \(p^2q^2\) with \(q\nmid(p^2-1)\) and \(p\nmid(q^2-1)\) are abelian
    683. Nilradical and infinitely many primes in \(F[x]\)
    684. The \(R/(p)\)-module structure on \(A/pA\)
    685. Irreducible polynomials over a field in which every element has a \(p\)th root are separable
    686. Fundamental theorem of Galois theory and \(\QQ(\zeta_{43})\)
    687. PIDs, and a UFD that is not a PID
    688. Quadratic extensions in characteristic not \(2\) are Galois
    689. A finite abelian group with at most \(n\) elements of order dividing \(n\) is cyclic
    690. Solving linear ODEs with constant coefficients using linear algebra
    691. \([\QQ(\sqrt{2}+\sqrt{3}):\QQ]\)
    692. \(x^{p^d}-x\) divides \(x^{p^n}-x\) iff \(d\) divides \(n\)
    693. A non-normal subgroup, and whether \(\mathrm{SO}(2)\normal\mathrm{SL}_2(\RR)\)
    694. Abelian groups are nilpotent
    695. Finite-dimensional division algebras over \(\RR\)
    696. Finite-dimensional representations of a compact Lie group are equivalent to unitary ones
    697. When a cyclic group is solvable
    698. Example of an Artinian ring
    699. Splitting fields of separable polynomials are Galois, with the Galois correspondence and compositum
    700. Classification of groups of order \(10\)
    701. Stabilizers need not be normal
    702. \(\Aut(\ZZ/p)\), \(\Aut((\ZZ/p)^n)\), and \(\Aut(\ZZ/n)\)
    703. Conjugacy classes of \(3\times 3\) rational matrices satisfying \(A^4-A^3-A+1=0\)
    704. Finite fields with \(p^n\) elements
    705. Galois group of \(x^3+6x+3\)
    706. Singular value decomposition
    707. Groups of order \(p^2 q\) have a nontrivial normal subgroup
    708. Every irreducible representation of \(G\) appears in a tensor power of a faithful finite-dimensional representation
    709. Whether real matrices conjugate over \(\CC\) are conjugate over \(\RR\)
    710. The center of \(S_3\) is trivial
    711. Galois groups of cubics
    712. Dimensions of the irreducible representations of \(D_n\), and \(\sum_i(\dim V_i)^2=|G|\)
    713. Elements \(g\) for which \(\chi(g)\) is real for every character \(\chi\)
    714. Smith normal form of a matrix over a PID
    715. Groups with nontrivial automorphisms
    716. The number and sizes of Jordan blocks
    717. A \(k\)-cycle as a product of transpositions
    718. Ideals, \(\spec\), \(\maxspec\), and radicals
    719. A finite abelian group is the product of its Sylow subgroups
    720. Cyclic extensions of prime order \(p\)
    721. Induced representations of \(G\) from a subgroup \(H\)
    722. Finite extensions are algebraic
    723. The compositum of Galois extensions \(E/k\) and \(F/k\) with \(E\cap F=k\) is Galois with \(G(EF/k)\cong G(E/k)\times G(F/k)\)
    724. The torsion submodule of a module over an integral domain
    725. Representations of a finite \(p\)-group on a vector space over a finite field of characteristic \(p\)
    726. Irreducible representations of \(\SO(2)\) and \(\SO(3)\)
    727. Groups of prime order are cyclic
    728. A nilpotent operator is diagonalizable iff it is zero
    729. Conjugacy classes in \(\mathrm{GL}_2(\CC)\)
    730. Matrix algebra
    731. Homomorphisms from \(S_n\) to a group of odd order
    732. A polynomial over \(\QQ\) with Galois group \(S_n\), in particular \(S_7\)
    733. Maschke’s theorem
    734. Example of a UFD that is not a Euclidean domain
    735. Jordan canonical form over a non-algebraically closed field
    736. No simple group of order \(pq^k\) when \(k\) is least with \(p\mid q^k-1\)
    737. The radical of an ideal is the intersection of the primes containing it
    738. An irreducible in \(\FF_p[x]\) divides \(x^{p^n}-x\) iff its degree divides \(n\)
    739. The stabilizer of a point under a group action is a subgroup
    740. An element of a commutative ring is noninvertible if and only if it lies in a maximal ideal
    741. In characteristic \(p\), an irreducible polynomial is \(g(x^{p^e})\) with \(g\) separable, and every root has multiplicity \(p^e\)
    742. The group algebra over \(\CC\) versus other fields, and when the characteristic divides \(|G|\)
    743. A cyclic permutation of a basis is diagonalizable with minimal polynomial \(x^n-1\)
    744. A basis of \(V\) yields a basis of \(V^{\oplus m}\)
    745. Intermediate fields of a \(D_8\)-extension with \(E/F\) and \(K/E\) Galois but \(K/F\) not Galois
    746. \(\Aut(S_n)=\Inn(S_n)\) for \(n\neq 6\)
    747. Group action, orbit, stabilizer, and fixed points
    748. Galois group of \(x^n-1\) over \(\QQ\)
    749. Regular representation
    750. The center of \(S_n\) is trivial for \(n\geq 4\)
    751. Nilpotent groups are solvable
    752. Galois group of \(x^2+9\)
    753. A \(p\)-group acting on a set of cardinality not divisible by \(p\) has a fixed point
    754. The cyclotomic extension \(\QQ(\zeta_{43})/\QQ\) has Galois group \(\ZZ_{42}\)
    755. Classification of groups of order \(4\), and which arise as Galois groups over \(\mathbb{Q}\)
    756. \(\Out(A_4)\) is nontrivial
    757. Maximal subgroups of a \(p\)-group have index \(p\)
    758. No simple group of order \(40\)
    759. Galois group of \(x^7-3\)
    760. Structure theorem for simple rings
    761. \(|\mathrm{GL}_2(\FF_q)|\) and its representations
    762. Subgroups of finitely generated free abelian groups and of free groups
    763. Lagrange’s theorem
    764. Character table of \(S_5\)
    765. \(x^{p^n}-x\) is the product of monic irreducibles in \(\FF_p[x]\) of degree dividing \(n\)
    766. Galois group of a finite field
    767. Classification of groups of order \(12\)
    768. A finite separable splitting field \(L/K\) has \([L:K]=|\gal(L/K)|\)
    769. \([A_n,A_n]=A_n\) for \(n\geq 5\)
    770. Field homomorphisms from \(\mathbb{Q}(\alpha)\) into \(\mathbb{C}\), \(\mathbb{R}\), and \(\overline{\mathbb{Q}}\) for a root \(\alpha\) of \(x^{17}-2\)
    771. \(\GF(p^d)\subseteq\GF(p^n)\) iff \(d\) divides \(n\)
    772. Eigenspaces of commuting matrices
    773. Schur’s lemma: endomorphisms of a simple module are zero or isomorphisms
    774. Whether an integral domain must be a PID if every finitely generated module is a direct sum of cyclic modules
    775. Invariant factors from the elementary divisors \((x-1)^3\), \((x-1)\), \((x^2+1)^4\), \((x^2+1)^2\), \((x^2+1)^2\), and \((x+2)\)
    776. An irreducible polynomial of degree \(p\) over \(\FF_p\) for each prime \(p\)
    777. Rank-nullity over division rings, and left inverses of rectangular matrices
    778. \([\QQ(\zeta+\zeta^{-1}):\QQ]=\varphi(n)/2\) for a primitive \(n\)th root of unity \(\zeta\)
    779. \([A_4,A_4]\cong\ZZ_2^2\)
    780. Unit quaternions and \(\mathrm{SO}(3)\)
    781. Stabilizer of a point in the unit disk under conformal automorphisms
    782. A single Jordan block of size \(6\) for \(T\) with \(p_T(x)=\chi_T(x)=x^6\)
    783. Whether \(L^1\) has a natural multiplication making it an algebra
    784. The ring of arithmetic functions under Dirichlet convolution
    785. Neither diagonalizable nor invertible implies the other
    786. Radical of an ideal
    787. The minimal polynomial over \(L\) divides the minimal polynomial over \(F\)
    788. Groups of order \(p^2\)
    789. The derived series, the commutator, and two theorems on derived series
    790. An ideal maximal among annihilators of nonzero elements of an \(R\)-module is prime
    791. Radical ideals in \(\ZZ\)
    792. Real division algebras
    793. Galois group \(D_6\) and intermediate fields of \(\QQ(\sqrt[3]{2},\sqrt{3},\zeta_3)/\QQ\)
    794. Abelianness of Galois groups is not transitive
    795. \(\langle 2,x\rangle\) is not principal in \(\ZZ[x]\)
    796. Structure theorem for finitely generated modules over a PID: invariant factors and elementary divisors
    797. Root bounds for polynomials over division rings
    798. Stabilizer subgroups of two points in a \(G\)-set are conjugate
    799. Proper subgroups of a group of order \(pq\) are cyclic
    800. Cayley-Hamilton theorem
    801. An ideal containing a unit is the whole ring
    802. Intermediate fields of \(\QQ(\sqrt{2},\sqrt{3})\) and \(\QQ(\sqrt{2}+\sqrt{3})\)
    803. Conjugacy classes in \(\mathrm{SL}_2(\RR)\)
    804. A matrix is diagonalizable iff all elementary divisors are linear
    805. Groups of order \(pqr\) have a normal Sylow subgroup
    806. Symmetry groups of the tetrahedron, cube, and icosahedron
    807. Every nonzero homomorphism from a field is injective, and whether the same holds for domains
    808. Kummer extension
    809. The Sylow theorems
    810. Character table of \(A_4\)
    811. Character orthogonality and detecting irreducible constituents
    812. Galois-group conditions implied by irreducibility of a polynomial, and by a root in the base field
    813. \(S_n\) is generated by transpositions
    814. \(IS\) is a submodule of \(A\)
    815. \(\Aut((\ZZ/p)^n)\) as a matrix group
    816. Algebraic closure
    817. When the Galois group of a polynomial is contained in \(A_n\)
    818. Quadratic extensions in characteristic 0 are Galois; stacked quadratics need not be
    819. Fundamental theorem of Galois theory, and the splitting field of \(x^5-2\)
    820. \(R^n/\im A\) is torsion iff \(\operatorname{rank} A=n\) iff the Smith form of \(A\) has \(n\) nonzero invariant factors
    821. Eigenvector \([0,1,0]^t\), rank, and diagonalizability of \(\begin{pmatrix}1&0&x\\0&1&0\\y&0&1\end{pmatrix}\)
    822. No simple group of order \(160\), and the structure of groups of that order
    823. The radical of an ideal is an ideal, and \(I(Z(I))=\sqrt{I}\)
    824. \(n_p=1\) when \(n_p\mid q<p\)
    825. Equality of the number of conjugacy classes and the number of irreducible representations of a finite group
    826. Maximal real subfield of \(\QQ(\zeta_n)\)
    827. Algebraically closed field
    828. A polynomial with Galois group \(\ZZ/3\ZZ\)
    829. \(4\times 4\) Jordan forms with minimal polynomial \((x-1)(x-2)^2\)
    830. Abelian groups of order 9, and groups of order 27

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