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Spam yourself with math PhD qualifying-exam questions.

UCLA Fall 2000, D2 Algebra, matrix similarity, diagonalization, rational canonical form

Describe all nondiagonalizable $4 \times 4$ matrices over $ℚ$ such that $𝐴^{5} + 𝐴^{2} = 0$ , up to similarity. Justify your answer.

UCLA Fall 2000, D3 Algebra, adjoint operators, normal operators, invariant subspaces, inner products

Let $𝑉$ be a complex vector space with positive-definite inner product $(\cdot, \cdot)$ , and let $𝑇 : 𝑉 \to 𝑉$ be a linear map. Recall that the adjoint $𝑇^{*}$ of $𝑇$ is defined by

(𝑇 (𝑥), 𝑦) = (𝑥, 𝑇^{*} (𝑦))

for all $𝑥, 𝑦 \in 𝑉$ , and that $𝑇$ is called normal if $𝑇$ and $𝑇^{*}$ commute. Suppose that $𝑇^{3} = 𝑇 𝑇^{*}$ . Let $𝑈$ be the kernel of $𝑇$ and $𝑊$ the orthogonal complement of $𝑈$ in $𝑉$ .

(a) Show that $𝑊$ is $𝑇^{*}$ -invariant.

(b) Show that $𝑈$ is $𝑇^{*}$ -invariant.

(d) Show that the restrictions of $𝑇$ and $𝑇^{*}$ to $𝑊$ commute.

(e) Show that $𝑇$ is normal.

UCLA Fall 2001, F1 Algebra, irreducible polynomials, field extensions, algebraic elements

Let $𝑓 (𝑥) = 𝑥^{3} - 2 𝑥 - 2$ .

(a) Show that $𝑓 (𝑥)$ is irreducible over $ℚ$ .

(b) Let $𝜃$ be a complex root of $𝑓 (𝑥)$ . Express $𝜃^{- 1}$ as a polynomial in $𝜃$ with coefficients in $ℚ$ .

UCLA Fall 2001, F2 Algebra, irreducible polynomials, cubic polynomials, rational roots

Let $𝑓 (𝑥) = 𝑥^{3} + 𝑛 𝑥 + 2$ where $𝑛$ is an integer. Determine the (infinitely many) values of $𝑛$ for which $𝑓$ is irreducible over $ℚ$ .

UCLA Fall 2001, F3 Algebra, Galois groups, radical extensions, matrix groups, cyclotomic fields

Let $𝐺$ be the Galois group of $𝑥^{𝑝} - 2$ over $ℚ$ where $𝑝$ is a prime. Show that $𝐺$ is isomorphic to the group of matrices of the form $(\begin{matrix} 𝑎 & 𝑏 \\ 0 & 1 \end{matrix}),$ where $𝑎 \in {(\frac{ℤ}{𝑝})}^{*}$ and $𝑏 \in \frac{ℤ}{𝑝}$ .

UCLA Fall 2001, G1 Algebra, finite groups, centers, conjugacy classes

Let $𝐺$ be a finite group whose center has index $𝑛$ . Show that every conjugacy class in $𝐺$ has at most $𝑛$ elements.

UCLA Fall 2001, G2 Algebra, group actions, stabilizers, symmetric groups, normal cores

Let $𝐺$ be a subgroup of $𝑆_{𝑛}$ that acts transitively on the set ${1, 2, \dots, 𝑛}$ . Let $𝐻$ be the stabilizer in $𝐺$ of an element $𝑥 \in {1, 2, \dots, 𝑛}$ . Prove that

⋂_{𝑔 \in 𝐺} 𝑔 𝐻 𝑔^{- 1} = {𝑒} .

UCLA Fall 2001, G3 Algebra, matrix groups, normal subgroups, finite fields, semidirect products

Let $𝐺$ be the group of matrices of the form $(\begin{matrix} 𝑎 & 𝑏 \\ 0 & 1 \end{matrix}),$ where $𝑎 \in {(\frac{ℤ}{𝑝})}^{*}$ and $𝑏 \in \frac{ℤ}{𝑝}$ . Describe all normal subgroups of $𝐺$ . Hint: find a convenient normal subgroup of order $𝑝$ .

UCLA Fall 2001, LA1 Algebra, linear operators, images, kernels, dimension

Let $𝑇$ be a linear operator on a finite-dimensional vector space $𝑉$ such that $Im (𝑇)$ and $Im (𝑇^{2})$ have the same dimension. Show that $ker (𝑇) \cap Im (𝑇) = 0$ .

UCLA Fall 2001, LA2 Algebra, matrix similarity, minimal polynomials, polynomial identities

Find all similarity classes of $4 \times 4$ matrices $𝐴$ over $ℚ$ such that $𝐴^{2} \neq \pm 𝐴$ and $𝐴^{2} \neq 𝐼$ but $𝐴^{3} = 𝐴$ ( $𝐼$ is the identity $4 \times 4$ matrix).

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