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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Spring 2026, 2 Algebra, group actions, conjugation, index

Let $𝐺$ be a finite group and let $𝐻 \leq 𝐺$ be a subgroup. Prove that the number of conjugates of $𝐻$ in $𝐺$ divides the index $[𝐺 : 𝐻]$ .

UC Irvine Spring 2026, 3 Algebra, symmetric group, normal subgroups, quotient groups

Prove that the symmetric group $𝑆_{4}$ has a normal subgroup $𝑁$ such that $𝑆_{4} / 𝑁 ≅ 𝑆_{3}$ .

UC Irvine Spring 2026, 4 Algebra, unique factorization domains, localization, principal ideal domains

Let $𝑅$ be a unique factorization domain, and let $𝑝 \in 𝑅$ be an irreducible element. Let $𝐾$ denote the field of fractions of $𝑅$ . Prove that the subring

𝑆 = {\frac{𝑎}{𝑏} \in 𝐾 ∣ 𝑎, 𝑏 \in 𝑅, 𝑝 ∤ 𝑏}

of $𝐾$ is a principal ideal domain with a single irreducible element up to associates.

UC Irvine Spring 2026, 5 Algebra, minimal polynomials, roots of unity, field extensions

Let $𝜁 = 𝑒^{2 𝜋 𝑖 / 7} \in ℂ$ be a primitive seventh root of unity. Find the minimal polynomial of $\sqrt[7]{5}$ over $ℚ (𝜁)$ .

UC Irvine Spring 2026, 6 Algebra, minimal polynomials, rational canonical form, matrix similarity

Show that there is a real $6 \times 6$ matrix whose minimal polynomial is $𝑥^{4} + 1$ and which is not similar to any matrix with rational entries.

UC Irvine Spring 2026, 7 Algebra, Galois groups, splitting fields, dihedral groups

Find the Galois group of the splitting field of the polynomial

𝑝 (𝑥) = 𝑥^{4} - 6 𝑥^{2} + 6

up to isomorphism. You may use without proof that $\sqrt{6} \notin ℚ (\sqrt{3 + \sqrt{3}})$ .

UC Irvine Spring 2026, 8 Algebra, finite fields, irreducible polynomials, field extensions

If $𝑞$ is a prime power, let $𝔽_{𝑞}$ denote the finite field of order $𝑞$ . Find all polynomials $𝑔 (𝑥) \in 𝔽_{2} [𝑥]$ such that

𝔽_{2} [𝑥] / (𝑔 (𝑥)) ≅ 𝔽_{16} .

UC Irvine Spring 2026, 9 Algebra, repeated roots, polynomial derivatives, greatest common divisors

Let $𝑛 \geq 1$ be an integer. Prove that the polynomial

𝑓 (𝑥) = 1 + \frac{𝑥}{1} + \frac{𝑥^{2}}{2!} + \dots + \frac{𝑥^{𝑛}}{𝑛!} \in ℚ [𝑥]

has no roots in $ℂ$ with multiplicity greater than $1$ .

Stanford Spring 2026, A.1 Algebra

Let $𝐴$ be a PID. Let $𝑀$ be a torsion $𝐴$ -module; i.e. there exists $𝑎 \in 𝐴$ such that $𝑎 𝑚 = 0$ for all $𝑚 \in 𝑀$ . Let $𝑛$ be a positive integer.

(a) Let $𝜙 : 𝐴^{𝑛} \to 𝑀$ be a surjective homomorphism of $𝐴$ -modules. Show that $ker 𝜙$ is a free $𝐴$ -module of rank $𝑛$ .

(b) Suppose $𝐴 = ℤ$ . If $𝑓_{1}, 𝑓_{2} : ℤ^{𝑛} \to 𝑀$ are surjective homomorphisms, show that there exist $𝑇 \in Aut (ℤ^{𝑛})$ and $𝑆 \in Aut (𝑀)$ , such that $𝑆 \circ 𝑓_{1} = 𝑓_{2} \circ 𝑇$ .

Stanford Spring 2026, A.2 Algebra

Let $𝑓 (𝑥) = 𝑥^{3} + 𝑥^{2} - 2 𝑥 - 1$ , regarded as a polynomial over a field $𝐾$ .

(a) Let $𝜁 \neq 0, 1$ be an element in an algebraic closure of $𝐾$ . Show that $𝜁 + 𝜁^{- 1}$ is a root of $𝑓 (𝑥)$ if and only if $𝜁^{7} = 1$ .

(b) Let $𝐾 = ℚ$ and $𝛼$ a root of $𝑓$ . Show that $ℚ (𝛼) / ℚ$ is Galois.

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