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Spam yourself with math PhD qualifying-exam questions.

UCLA Fall 2001, LA3 Algebra, bilinear forms, alternating forms, skew-symmetric forms, characteristic two

Let $𝑉$ be a vector space over a field $𝑘$ . A bilinear form $𝑓 : 𝑉 \times 𝑉 \to 𝑘$ is called skew-symmetric if $𝑓 (𝑢, 𝑣) = - 𝑓 (𝑣, 𝑢)$ for all $𝑣, 𝑢 \in 𝑉$ and is called alternating if $𝑓 (𝑣, 𝑣) = 0$ for all $𝑣 \in 𝑉$ .

(a) Prove that every alternating form is skew-symmetric.

(b) Give an example of a skew-symmetric form which is not alternating. Hint: choose $𝑘$ of characteristic $2$ .

UCLA Fall 2001, R1 Algebra, commutative rings, ideals, free modules, zero divisors

Let $𝑅$ be a commutative ring, $𝐼 \subset 𝑅$ a nonzero ideal. Prove that if $𝐼$ is a free $𝑅$ -module then $𝐼 = 𝑎 𝑅$ for an element $𝑎 \in 𝑅$ which is not a zero divisor in $𝑅$ . Hint: consider the rank of $𝐼$ .

UCLA Fall 2001, R2 Algebra, prime ideals, maximal ideals, commutative rings

(a) Give an example of a prime ideal in a commutative ring that is not maximal.

(b) Let $𝑅$ be a commutative ring with identity. Suppose for every element $𝑥 \in 𝑅$ there exists an integer $𝑛 = 𝑛 (𝑥) > 1$ such that $𝑥^{𝑛} = 𝑥$ . Show that every prime ideal in $𝑅$ is maximal.

UCLA Fall 2001, R3 Algebra, polynomial rings, units, nilpotent elements, integral domains

Let $𝑅$ be a ring.

(a) Prove that if $𝑎$ is a nilpotent element in a ring $𝑅$ with identity, then the element $1 + 𝑎$ is invertible.

In the next two parts, let $𝑓 (𝑋) = 𝑎_{0} + 𝑎_{1} 𝑋 + \dots + 𝑎_{𝑛} 𝑋^{𝑛}$ be a polynomial in $𝑅 [𝑋]$ of degree $𝑛$ , that is, $𝑎_{𝑛} \neq 0$ .

(b) Show that if $𝑅$ is an integral domain, then $𝑓 (𝑋)$ is invertible in $𝑅 [𝑋]$ if and only if $𝑛 = 0$ .

(c) Show that if $𝑅$ is a commutative ring, $𝑓 (𝑋)$ is invertible in $𝑅 [𝑋]$ if and only if $𝑎_{0}$ is invertible and $𝑎_{𝑖}$ is nilpotent in $𝑅$ for every $𝑖 \geq 1$ .

UCLA Fall 2002, 1 Algebra, field extensions, minimal polynomials, radicals

(a) Determine the minimal polynomial of $𝑢 = \sqrt{3 + 2 \sqrt{2}}$ over $ℚ$ .

(b) Determine the minimal polynomial of $𝑢^{- 1}$ over $ℚ$ .

UCLA Fall 2002, 1 Algebra, free abelian groups, finite index, quotient groups

(a) Let $𝐴$ be a free abelian group of rank $𝑛$ . If $𝐻$ is a subgroup of $𝐴$ , show that $𝐻$ is free abelian of rank $𝑛$ if and only if $𝐴 / 𝐻$ is finite.

UCLA Fall 2002, 1 Algebra, linear operators, images, kernels

Let $𝑇$ be a linear operator on a finite-dimensional vector space $𝑉$ such that $im 𝑇 = im 𝑇^{2}$ . Prove that $ker 𝑇 = ker 𝑇^{2}$ . (Here $im$ and $ker$ refer to the image and kernel of $𝑇$ .)

UCLA Fall 2002, 1 Algebra, commutative rings, maximal ideals, zero divisors, polynomial rings

Let $𝑅$ be a commutative ring with $1$ , and let $𝑆 = 𝑅 [𝑥]$ be the polynomial ring in one variable. Suppose $𝑀$ is a maximal ideal of $𝑆$ . Prove that $𝑀$ cannot consist entirely of zero divisors.

Hint: You may want to distinguish the cases $𝑥 \in 𝑀$ or $𝑥 \notin 𝑀$ .

UCLA Fall 2002, 2 Algebra, splitting fields, Galois groups, Galois correspondence

(a) Let $𝐹$ be the field generated by the roots of the polynomial $𝑋^{6} + 3$ over $ℚ$ . Determine the Galois group of $𝐹 / ℚ$ .

(b) Describe all subfields of $𝐹$ .

UCLA Fall 2002, 2 Algebra, finite groups, normal subgroups, Sylow theory

Let $𝐺$ be a finite group of order $108$ . Show that $𝐺$ has a normal subgroup of order $9$ or $27$ .

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