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Spam yourself with math PhD qualifying-exam questions.

UCLA Fall 2000, A1 Algebra, dihedral groups, subgroups, index

Let $𝐷_{2 𝑛}$ be the dihedral group of order $2 𝑛$ , with $𝑛 > 1$ . Determine the number of subgroups of $𝐷_{2 𝑛}$ of index $2$ , and justify your answer.

UCLA Fall 2000, A2 Algebra, finite groups, normal subgroups, p-groups, homomorphisms

A group of order a power of a prime $𝑝$ is called a $𝑝$ -group. Let $𝐺$ be a finite group. Prove that, for any given prime $𝑝$ , there exists a unique normal subgroup $𝑁$ of $𝐺$ such that

(i) $𝐺 / 𝑁$ is a $𝑝$ -group, and

(ii) any homomorphism $𝜋$ of $𝐺$ into a $𝑝$ -group is trivial on $𝑁$ (that is, $𝜋 (𝑁) = 1$ ).

UCLA Fall 2000, A3 Algebra, finite groups, cyclic groups, subgroup orders

Let $𝐺$ be a finite group of order $𝑛$ . Suppose that $𝐺$ has a unique subgroup of order $𝑑$ for each positive divisor $𝑑$ of $𝑛$ . Prove that $𝐺$ is a cyclic group.

UCLA Fall 2000, B1 Algebra, modules, Noetherian modules, Artinian modules, direct sums

Let $𝑀$ be a module over a commutative ring $𝐴$ . If every increasing (respectively, decreasing) sequence of $𝐴$ -submodules of $𝑀$ terminates after finitely many steps, the $𝐴$ -module $𝑀$ is called Noetherian (respectively, Artinian).

(a) Prove that the $ℤ$ -module $ℤ$ is Noetherian and non-Artinian.

(b) Prove that the $ℤ$ -module

⋃_{𝑛 = 1}^{\infty} (𝑝^{- 𝑛} ℤ / ℤ)

is Artinian and non-Noetherian.

UCLA Fall 2000, B2 Algebra, prime ideals, maximal ideals, nilpotent elements

Let $𝐴$ be a commutative ring with identity. Suppose that $𝑎 \in 𝐴$ is not nilpotent (that is, $𝑎^{𝑛} \neq 0$ for all $𝑛 > 0$ ).

(a) Prove that there exists a prime ideal $𝑃 \subset 𝐴$ such that $𝑎 \notin 𝑃$ .

(b) Give an example of a ring $𝐴$ and a non-nilpotent $𝑎 \in 𝐴$ such that $𝑎$ is contained in $𝑀$ for all maximal ideals $𝑀 \subset 𝐴$ . Justify your example.

UCLA Fall 2000, B3 Algebra, Noetherian rings, prime ideals, nilpotent ideals, product rings

Let $𝐴$ be a commutative Noetherian ring with identity $1 \neq 0$ . Write $𝑋 (𝔞)$ for the set of prime ideals of $𝐴$ containing a given ideal $𝔞$ . Suppose that

𝑋 (0) = 𝑋 (𝔞) \cup 𝑋 (𝔟)

and $𝑋 (𝔞) \cap 𝑋 (𝔟) = \emptyset$ for two ideals $𝔞$ and $𝔟$ . Prove the following facts:

(a) $𝐴 = 𝔞 + 𝔟$ ;

(b) $𝔞 \cap 𝔟 = 𝔞 𝔟$ ;

(d) there exists a positive integer $𝑛$ such that $𝐴$ is isomorphic to the product ring

(𝐴 / 𝔞^{𝑛}) \times (𝐴 / 𝔟^{𝑛}) .

UCLA Fall 2000, C1 Algebra, field extensions, minimal polynomials, radicals

Find the minimal polynomial of $\sqrt{2} + \sqrt[3]{3}$ over the field of rational numbers $ℚ$ .

UCLA Fall 2000, C2 Algebra, finite fields, roots of unity, extension degrees

Let $𝑎$ be a primitive 16-th root root of unity over the field $𝐹$ . Determine the dimension $[𝐹 (𝑎) : 𝐹]$ when $𝐹$ is:

(a) the field of $9$ elements;

(b) the field of $7$ elements;

(d) What can you say in the case $𝑝 = 2$ ?

UCLA Fall 2000, C3 Algebra, field extensions, simple extensions, intermediate fields, positive characteristic

Let $𝐹$ be an infinite field of characteristic $𝑝 > 0$ . Recall that a finite-dimensional extension $𝐿 / 𝐹$ is said to be simple if $𝐿 = 𝐹 (𝑢)$ for some element $𝑢 \in 𝐿$ .

(a) Suppose $𝐿 / 𝐹$ has a finite number of intermediate fields. Show that $𝐿 / 𝐹$ is simple.

(b) Let $𝐾$ be an intermediate field $𝐹 \subset 𝐾 \subset 𝐿$ , and suppose that $𝐿 = 𝐹 (𝑢)$ , with

𝑥^{𝑟} + 𝑎_{1} 𝑥^{𝑟 - 1} + \dots + 𝑎_{𝑟}

the monic irreducible polynomial of $𝑢$ over $𝐾$ , where $𝑎_{𝑖} \in 𝐾$ . Show that $𝐾 = 𝐹 (𝑎_{1}, 𝑎_{2}, \dots, 𝑎_{𝑟})$ .

(d) Let $𝐸 = 𝐹 (𝑥, 𝑦)$ , where $𝑥$ and $𝑦$ are indeterminates, and set

𝑀 = 𝐸 (𝑥^{\frac{1}{𝑝}}, 𝑦^{\frac{1}{𝑝}}) .

Show that $𝑀 / 𝐸$ has an infinite number of intermediate fields.

UCLA Fall 2000, D1 Algebra, linear operators, characteristic polynomials, diagonalization

Let $𝑉$ be a finite-dimensional vector space over a field $𝐹$ and $𝑇 : 𝑉 \to 𝑉$ a linear map whose characteristic polynomial has distinct roots, all in $𝐹$ . Show that $𝑇$ is diagonalizable.

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