60 problems.
Ph.D. qualifying exam: algebra, June 11, 2015 (pp. 1–3)
- A group of order \(150\) has a normal subgroup of order \(5\), \(10\), \(15\), or \(25\)
- \(A/A[p]\) versus \(A^p\), and \(A/A^p \cong A[p]\) when \(A\) is finite
- The identity bijection \(J\rtimes_\varphi L\to J\times L\) is an isomorphism exactly for the trivial action
- A proper subgroup of a finite \(p\)-group is proper in its normalizer
- Rational and Jordan canonical forms of a \(3\times 3\) real matrix; abelian groups of order \(360\)
- Direct summands are pure; a pure submodule of a finitely generated torsion module over a PID is a direct summand
- Prime and maximal ideals \((x^2+1)\), \((6,x)\), and \((y^2-x^3)\)
- Whether \(\ZZ[3i]\) is a UFD
- Splitting field of \(X^4-12\) over \(\mathbb{Q}\): explicit description, degree, and Galois group
- Galois group of irreducible \(X^p-a\) over \(\mathbb{Q}\) is the group of maps \(y\mapsto ky+l\) on \(\mathbb{Z}/p\mathbb{Z}\) with \(k\neq 0\)
- When \(X^{p^n}-X+1\) is irreducible over \(\mathbb{F}_p\)
- At most two \(F\)-conjugates of \(L\) when \([K:F]=2\) and \(L/K\) is finite Galois
Algebra Preliminary Examination, Summer 2014 (pp. 4–5)
- Polynomial PIDs, cyclic subgroups of \(S_8\), and two counterexamples
- A group of order \(84\) with \(28\) Sylow \(3\)-subgroups has a normal Sylow \(7\)-subgroup \(K\) with \(|Z_G(K)|=28\)
- Units of a local ring, nonarchimedean valuation rings, and \(R/M^n\)
- \((R/I)\otimes_R M\cong M/IM\)
- \(\mathbb{Q}[x]\)-modules from order-\(6\) matrices in \(M_4(\mathbb{Q})\), and groups of order \(24\)
- Galois group of the splitting field of \(x^3+2\) over \(\mathbb{Q}\), and the subextension \(\mathbb{Q}(\zeta_3)\)
Wesleyan University PhD Qualifying Exam, Algebra, July 10, 2013 (pp. 6–8)
- Sylow’s theorem, and groups of order \(pq\) are cyclic when \(p \nmid q-1\)
- A subgroup of index \(n\) contains a normal subgroup \(K\) with \([G:K]\) dividing \(n!\)
- A subgroup of index equal to the smallest prime dividing \(|G|\) is normal
- A counterexample to the converse of Lagrange’s theorem
- Definition of an ideal, the join \(I\vee J\), the product \(IJ\subseteq I\cap J\), and whether every ideal contained in \(I\) and \(J\) lies in \(IJ\)
- Abelian groups of order \(60\), and \(\mathbb{Q}[x]\)-modules of dimension \(4\) annihilated by \(x^8-1\)
- Isomorphism type of \(\mathbb{Z}^3/\langle(0,4,2),(-1,-4,-1),(0,0,-2)\rangle\)
- Gcd ideals and compatible congruences in a PID
- Splitting field of \(x^4-6\) over \(\QQ\): degree, Galois group, and the number of subfields of each degree, Galois or not
- Finite subgroups of field automorphism groups, and of finite-field automorphism groups
- Algebraic elements form a subfield, and the minimal polynomial of \(2\sqrt{2}-\sqrt{3}\)
- Fixed field of \(\sigma(x)=1-1/x\) acting on \(K(x)\)
ALGEBRA QUALIFYING EXAM, 13 JUNE 2012 (pp. 9–11)
- Sylow subgroup counts and exclusion of proper subgroups of index at most four
- A nonabelian group with all proper subgroups normal, abelian groups with non-normal subgroups, groups of order \(p^2\), missing subgroup orders, infinite torsion groups, and non-isomorphic subgroups of order \(p^k\)
- Sylow subgroups of \(A_5\), their normalizers, and subgroups of orders \(6\), \(10\), and \(15\)
- A group of order \(30\) has a subgroup of order \(15\)
- Prime but not maximal ideals, finitely generated torsion-free modules that are not free, torsion-free modules over a PID that are not free, Euclidean domains that are not UFDs, UFDs that are not PIDs, and non-Noetherian domains
- Finitely generated modules over a PID, groups of order \(72\), and rational canonical form
- If \(I+J=R\) then \(IJ = I \cap J\) and \(R/(IJ) \cong R/I \oplus R/J\)
- The quotients \(\mathbb{Z}[x]/(2,x^2+3x-1)\) and \(\mathbb{Z}[x]/(2,x^3+3x^2-x+4)\)
- Splitting field of \(x^3-5\) over \(\mathbb{Q}\): degree, Galois group, subfields, and which are Galois
- Examples and non-examples in Galois theory
- \((\mathbb{Z}/2\mathbb{Z})[y]/(y^4+y^3+y^2+y+1)\) is a field over which \(x^4+x+1\) and \(x^2+x+1\) split and \(x^3+x+1\) is irreducible
- Minimal polynomial degrees in a Galois extension with group \(A_n\)
Algebra Preliminary Examination, Summer 2011 (p. 12)
- \(\mathbb{Q}(\sqrt[3]{2})\) in cyclotomic extensions, splitting fields over finite fields, and homomorphisms from \(S_4\)
- A nonzero commutative ring with every proper principal ideal prime is a field
- Similarity classes of order-\(8\) matrices in \(M_n(\mathbb{Q})\) for \(n=3,4,5\)
- Every group of order \(15\) is cyclic, and every group of order \(255\) is cyclic
- Free \(R\)-modules of rank at most \(n\) in an \(n\)-dimensional vector space over a PID
- The splitting field of \((x^p-2)(x^q-2)\) over \(\mathbb{Q}\)
Algebra Qualifying Exam, June 16, 2010 (pp. 13–15)
- The Sylow theorems
- A finite abelian group is cyclic iff it has a unique subgroup of each order dividing \(|A|\)
- Frattini’s argument: \(G=KN_G(P)\) for a Sylow \(p\)-subgroup \(P\) of a normal subgroup \(K\)
- Nonabelian groups of order \(15\), \(20\), and \(21\), or a proof that none exist
- Examples of a PID, a UFD, a torsion-free module, and a vanishing tensor product
- In a local ring every element is a unit or lies in the unique maximal ideal
- \(A/mA \cong A \otimes_{\mathbb{Z}} (\mathbb{Z}/m\mathbb{Z})\)
- Rational canonical form of a \(3\times 3\) matrix over \(\mathbb{Q}\), and a non-similar matrix with the same characteristic polynomial
- Intermediate fields of a Galois extension with group \(D_8\)
- A Galois tower that is not Galois, and a finite-dimensional inseparable extension
- Irreducibility of a degree-\(4\) polynomial over \(\mathbb{F}_p\) via \(\gcd(f,x^{p^2}-x)\), and its Galois group
- Whether \(\mathbb{Q}(\sqrt[3]{2})\) and \(\mathbb{Q}(\sqrt{2}+\sqrt{5})\) are splitting fields over \(\mathbb{Q}\)