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    University of Washington algebra prelims, 2005–2018

    SRC-UW-ALGEBRA-2005-2018

    Area
    Algebra
    Status
    draft
    Source
    Contents

    80 problems.

    2018

    1. No finite group is the union of conjugates of a proper subgroup
    2. Groups of order 18
    3. Degree of \(\QQ(7^{1/5},5^{1/4})\) over \(\QQ\)
    4. Galois group of \(\QQ(\sqrt{2+\sqrt{2}})/\QQ\)
    5. A square matrix is similar to its transpose
    6. Equivalence of irreducible representations versus equality of determinants

    2017

    1. Hilbert Basis Theorem for \(R[x]\) and \(R[[x]]\)
    2. Classification of finite fields
    3. Kernels of finite free presentations of finitely presented modules are finitely generated
    4. Groups of order 57
    5. A finite simple group has no 2-dimensional irreducible complex representation

    2016

    1. A finite simple group with all proper subgroups abelian is cyclic of prime order
    2. Galois group of \(x^5-5a^4x+a\) over \(\QQ\)
    3. An outer automorphism of \(S_6\) from a transitive embedding \(S_5\hookrightarrow S_6\)
    4. Equivalence of \(\mathrm{Ann}(M)\not\subset\mathfrak{p}\), \(M_{\mathfrak{p}}=0\), and \(M\otimes_A k(\mathfrak{p})=0\) for finitely generated modules
    5. \(\CC[x,y]/(y^2-(x-1)^3-(x-1)^2)\) is a domain, its real points, and its integral closure
    6. A projective resolution of \(k\simeq R/(x,y)\) over \(R=k[x,y]\) and \(\mathrm{Tor}_i^R((x,y),k)\)

    2015

    1. A degree-5 irreducible over Z/2, the field of 32 elements, and a matrix of order 31
    2. Minimal polynomial of \(\sqrt{2}+\sqrt{3}\) over \(\QQ\)
    3. Units of \(R[x]\) for reduced \(R\), and units of \(\mathbb{Z}_4[x]\)
    4. At most one nonabelian group of order \(pq\), and the pairs of distinct primes with none
    5. The number of intermediate fields of a degree-\(4\) Galois extension, and of a degree-\(4\) separable extension
    6. Derivations of \(\CC[x]\) and simplicity of \(\CC[x]\) as a Weyl module
    7. Center, linear characters, and irreducible representations of a nonabelian group of order \(p^3\)

    2014

    1. A group with a subgroup of index \(n\) has a normal subgroup \(N\) with \(n\leq[G:N]\leq n!\), and there is no simple group of order \(36\)
    2. \(x^p-t\) is irreducible over \(\mathbb{F}_p(t)\) and splits as a \(p\)-th power in its splitting field
    3. Recall that a ring \(A\) is called if it admits a direct
    4. Rings with \(a^2=a\): Jacobson radical, characteristic, commutativity, and finite rings isomorphic to \((\ZZ/2\ZZ)^d\)
    5. \(\Gal(\overline{\mathbb{F}_p}/\mathbb{F}_p)\) has no nontrivial finite subgroups
    6. Rational irreps of \(C_p\) of dimensions \(1\) and \(p-1\), and abelian groups whose rational representations are absolutely irreducible
    7. Class sums form a \(\mathbb{Z}\)-basis of the center of \(\mathbb{Z}[G]\), and \(|C|\chi_\pi(C)/\dim V\) is an algebraic integer

    2013

    1. A finitely generated group has finitely many subgroups of index \(n\), and the possible numbers of index-\(p\) subgroups of a finitely generated abelian group
    2. A group of order \(2013=3\cdot 11\cdot 61\) has a cyclic normal subgroup of index \(3\), and \(11\) divides \(|Z(G)|\)
    3. Unique Galois quartic \(\QQ(\sqrt{d_1},\sqrt{d_2})\) in a nonabelian Galois octic \(K/\QQ\); \(d_1,d_2>0\) when \(G=Q_8\)
    4. \((5,x^2+2)\) is prime in \(\mathbb{Z}[x,y]\) with infinitely many maximal extensions

    2012

    1. Groups of order 2012
    2. The dihedral presentation \(G_n=\langle a,b\mid a^2=b^2=(ab)^n=1\rangle\): order and irreducible representations of \(G_4\)
    3. Galois groups of \(x^4+4x^2+1\) and \(x^4+4x^2-5\) over \(\QQ\)
    4. Kernel of a map of finitely generated free modules over a PID is a direct summand; the image need not be

    2011

    1. No simple group of order \(30\), and simple groups of order \(60\) are isomorphic to \(A_5\)
    2. Galois group and intermediate fields of \(\mathbb Q(\zeta_{12})/\mathbb Q\)
    3. Whether \(\mathbb{Z}[x]/(x^2+x+1)\) is Noetherian, Artinian, or an integrally closed domain
    4. The nilradical is the intersection of all primes; \(N(\mathbb{Z}/(m))\) and \(N(\mathbb{C}[x,y]/(y^2-x^3))\)
    5. Irreducible representations of \(\mathrm{GL}_2(\mathbb{F}_2)\), and reducibility of the adjoint representation in order \(12\)
    6. A commutative integral domain is a field if and only if it is semisimple if and only if every module is projective

    2010

    1. Order of \(\GL_2(\FF_p)\) and its subgroups of order \(p\)
    2. Length of a module, composition series, and additivity of length along exact sequences
    3. The sum of group elements is central and nilpotent in \(\mathbb{F}[G]\) when \(\mathrm{char}\,\mathbb{F}=p\) and \(|G|=p^n\), so \(\mathbb{F}[G]\) is not semisimple
    4. \(\ZZ[x]/(f)\) is a finitely generated \(\ZZ\)-module if and only if the leading coefficient of \(f\) is \(\pm 1\)
    5. Units and maximal ideals of \(C[0,1]\)
    6. An algebraic extension of a characteristic-zero field in which every polynomial over \(F\) has a root is algebraically closed
    7. Character table of the nonabelian group of order \(21\)

    2009

    1. Groups of order 2009 and their intermediate subgroups
    2. The valuation ring of a discrete valuation, and the \(p\)-adic valuation on \(\mathbb{Q}\)
    3. Quadratic extensions are \(F(\sqrt{D})\) when \(\operatorname{char} F\neq 2\), and \([F(\sqrt{D_1},\sqrt{D_2}):F]\)
    4. A root extension of an irreducible polynomial: existence, basis, and \(\theta^{-1}\); irreducibility of \(x^3+9x+6\) and the inverse of \(1+\theta\)
    5. Chain conditions force injective or surjective module endomorphisms to be automorphisms
    6. Schur’s lemma: \(\operatorname{Hom}_{kG}(V,V)=k\) for an irreducible representation over an algebraically closed field

    2008

    1. An irreducible quintic over \(\mathbb{Q}\) with three real roots is not solvable by radicals, and a quadratic Galois subfield of its splitting field
    2. A finite field has irreducible polynomials of every positive degree
    3. Order of \(\GL_n(\FF_q)\)
    4. Two-generated modules over a PID are \(R/(a)\oplus R/(b)\), and when two such sums are isomorphic
    5. The subgroup of \(\GL_3(\CC)\) generated by permutation matrices of \(S_3\) and \(\mathrm{diag}(i,1,1)\)
    6. Irreducible \(\CC G\)-modules have dimension at most \(\sqrt{|G|}\), with an example of dimension \(\lfloor\sqrt{|G|}\rfloor\)
    7. A square matrix is similar to its transpose

    2007

    1. An irreducible of degree \(7\) over \(K\) with no zeros in a Galois extension \(L\) remains irreducible in \(L[x]\)
    2. Galois group of an irreducible of degree \(n\) embeds in \(S_n\); examples with \(G=S_n\), and the cubic case
    3. Exactly two groups of order 21

    2006

    1. Irreducibility, splitting field, and Galois group of \(x^n-z\) over \(\mathbb{Q}(z)\)
    2. Groups of order \(pqr\) are not simple; simplicity of groups of order \(12p\) for \(p=5,7,11\)
    3. The Galois module \(K\) is the regular representation, and cyclic Kummer generators
    4. The group of upper-triangular unipotent \(3\times 3\) matrices over \(\mathbb{F}_p\) is nonabelian, \(g^p=I\) for \(p\) odd, and \(D_8\) versus the quaternionic group for \(p=2\)
    5. Minimal faithful permutation degrees of the groups of order \(8\)

    2005

    1. For any group \(G\) we define \(\Omega(G)\) to be the image of the group
    2. Order of \(\mathrm{GL}_3(\mathbb{F}_2)\) and the number of elements of order \(7\)
    3. The kernel of a surjection \(\ZZ^m\to G\) onto a finite abelian group is isomorphic to \(\ZZ^m\), with \(|\det A|=|G|\)
    4. Units and maximal ideals of \(C[0,1]\)
    5. Inverse images of primes are prime, the nilradical is contained in every prime, and \(\mathrm{Spec}(R/N)\to\mathrm{Spec}(R)\) is bijective
    6. Irreducibility, degree, and Galois group of \(x^{10}+x^5+1\) over \(\QQ\)
    7. Generators of \(\mathbb F_{p^6}\) inside \(\mathbb F_{p^n}\)

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