Algebra

Let $V$ be a real vector space of dimension at least $3$ and let $T \in \operatorname {End}_{\mathbb {R}}(V)$. Prove that there is a non-zero subspace $W$ of $V$, $W \neq V$, such that $T(W) \subseteq W$.

Let $V$ be a finite-dimensional vector space over a field $K$. Let $S$ be a linear transformation of $V$ into itself. Let $W$ be an invariant subspace of $V$ (that is, $SW \subseteq W$). Let $m(t)$, $m_1(t)$, and $m_2(t)$ be the minimal polynomial of $S$ as linear transformation of $V$, $W$ and $V/W$ respectively.

Let $V$ be a finite-dimensional vector space over $\mathbb {R}$ and $T : V \to V$ be a linear transformation such that (a) the minimal polynomial of $T$ is irreducible and (b) there exists a vector $v \in V$ such that $\left\{ T^i v \mid i \geq 0\right\}$ spans $V$. Show that $V$ does not have a proper $T$-invariant subspace.

Let $A$ be an $n \times n$ matrix with entries in $\mathbb {C}$. Show that $A$ has $n$ distinct eigenvalues in $\mathbb {C}$ if and only if $A$ commutes with no nonzero nilpotent matrix.

Suppose $V$ is a finite-dimensional vector space over a field $F$, $T : V \to V$ a linear map such that the minimal polynomial of $T$ coincides with the characteristic polynomial, which is the square of an irreducible polynomial in $F[\lambda ]$. Show that if $u$, $v$ and $w$ are any three non-zero vectors in $V$, then at least two of the three subspaces spanned by the sets $\left\{ T^i u\right\}_{i \geq 0}$, $\left\{ T^i v\right\}_{i \geq 0}$ and $\left\{ T^i w\right\}_{i \geq 0}$ coincide.

Let $V$ be a finite-dimensional vector space over $\mathbb {C}$ with basis $\left\{ v_1, \ldots , v_n\right\}$. Let $\sigma$ be a permutation on $\left\{ v_1, \ldots , v_n\right\}$ and thus induce a linear transformation $A$ on $V$. Show that $A$ is diagonalizable.

Let $V$ be a finite-dimensional vector space over the field of rational numbers. Suppose $T$ is a non-singular linear transformation of $V$ such that $T^{-1} = T^2 + T$. Prove that $3$ divides the dimension of $V$, and prove that if $\dim V = 3$, then all such $T$'s are similar.

Let $\mathbb {R}_q$ be a finite field with $q = p^n$ elements where $p$ is a prime. Let $\Pi : \mathbb {R}_q \to \mathbb {R}_q$ be the Frobenius automorphism $\Pi (x) = x^p$. Prove that $\Pi$ considered as a linear map over $\mathbb {R}_p$ is diagonalizable if and only if $n$ divides $p - 1$.

Let $A(t)$ be a non-singular matrix whose elements are differentiable functions of real variable $t$. Let $A'(t)$ denote the matrix formed by the derivatives of the elements. Show that the derivative of the determinant $\det A$ satisfies

Let $V$ be the vector space of polynomials $p(x) = a + bx + cx^2$ with real coefficients $a$, $b$, and $c$. Define an inner product on $V$ by

Suppose that $A$ is an $n \times n$ matrix and that

Suppose that all the entries of $A$, $x$, and $y$ are real.

Let $A$ be an $n \times n$ real matrix with distinct (possibly complex) eigenvalues, $\lambda_1, \lambda_2, \ldots , \lambda_n$, and corresponding eigenvectors $v_1, v_2, \ldots , v_n$. Assume that $\lambda_1 = 1$ and that $\left|\lambda_j\right| < 1$ for $2 \leq j \leq n$. Prove that $\lim_{n \to \infty } A^n v$ exists. Define $T : \mathbb {C}^n \to \mathbb {C}^n$ by $T(v) = \lim_{n \to \infty } A^n v$. Find the dimensions of the kernel and image of $T$ and give a basis for both.

Let $A$ be a matrix. Define

For

express $\sin (A)$ in closed form.

Let $A$ be the $9 \times 9$ tridiagonal matrix

All entries not shown are zero.

Let $T$ be a real symmetric, positive-definite, $n \times n$ matrix with distinct eigenvalues $\lambda_1 > \lambda_2 > \cdots > \lambda_n$. Show that

where $V$ ranges over all two-dimensional subspaces of $\mathbb {R}^n$ and $\left|x\right|$ is the Euclidean norm of $x$.

For any $n \times n$ matrix $P$, consider the sum

Let $G$ be a $p$-group and $G \times V \to V$ be a linear action of $G$ on a finite vector space over a finite field $F_{p^n}$. Using Sylow theory, prove that there exist a basis of $V$ in which all the transformations $v \to \sigma \cdot v$ $(\sigma \in G)$ are unipotent matrices.

Let $S$ be an endomorphism of a finite-dimensional vector space $V$ over a field $F$ whose characteristic polynomial is not equal to its minimal polynomial. Show that there is an endomorphism $T$ of $V$ so that $T$ commutes with $S$ but $T$ is not a polynomial in $S$. ($T$ is a polynomial in $S$ if $T = a_0 I + a_1 S + a_2 S^2 + \cdots + a_k S^k$ for some $k$ and $a_i \in F$.)

Let $A$ be an $n \times n$ real matrix, all of whose (complex) eigenvalues are real and positive. Show that for any integer $m \geq 1$, there exists at least one $n \times n$ real matrix $B$ with $B^m = A$.

Let $V$ be a finite-dimensional vector space over an algebraically closed field $K$, and let $T : V \to V$ be a linear transformation.

Let $G$ be the group of real $2 \times 2$ matrices, of determinant one. Describe the set of conjugacy classes of elements of $G$.

Let $G = GL_n(F_q)$, the group of invertible $n \times n$ matrices over the finite field $F_q$ with $q = p^r$, $p$ a prime, $U = U_n(F_q)$, the subgroup of upper triangular matrices with $1$'s on the diagonal.

Suppose that $H$ and $K$ are normal subgroups of a finite group $G$ and that $G/H \simeq K$.

Prove that if $G$ is a finitely generated infinite group then for each positive integer $n$, $G$ has only finitely many subgroups of index $n$.

Show that $G$ is solvable but not nilpotent.

Let $p$ be a prime and let $V$ be an $n$-dimensional vector space over $F_p$. Let $G = GL_n(V)$. Recall that $\left|G\right| = (p^n - 1)(p^n - p) \cdots (p^n - p^{n-1})$.

Let $p$ and $q$ be distinct prime numbers. Let $G$ be a group of order $p^3 \cdot q$ such that its commutator subgroup $K$ is of order $q$. Let $H$ be a $p$-Sylow subgroup of $G$.

Let $A$ be the abelian group on generators $e$, $f$, and $g$, subject to the relations

Give a decomposition of $A$ as a direct sum of cyclic groups of prime order or infinite order.

Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers, i.e., that there exists an $n \times n$ matrix $N$ with rational entries so that $MN$ equals the $n \times n$ identity matrix. View $M$ as an endomorphism of $\mathbb {Z}^n$.

Let $D = \mathbb {Z}[\xi ]$ with $\xi = \frac{-1+\sqrt{-3}}{2}$. Calculate the order of the additive group $G = D^2/K$ where $K$ is the $D$-submodule of $D^2$ generated by $(2\xi +1, \xi -1)$, $(\xi +2, \xi -4)$ and $(21, 21)$. Then express $G$ as a product of cyclic groups.

Prove that $SL(2, \mathbb {Z})$ is generated by $\begin{pmatrix} 1 & \pm 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ where $SL(2, \mathbb {Z})$ is the group of $2 \times 2$ matrices with integral coefficients and determinant $= 1$.

Let $G$ be a finite group, $K$ a normal subgroup of $G$ and $P$ a Sylow subgroup of $K$. If $N$ is the normalizer of $P$ in $G$ (i.e., $N = \left\{ g \in G \mid gPg^{-1} = P\right\}$) then show that $G = K \cdot N$.

Let $P$ be a Sylow subgroup of a finite group $G$. Let $N = \left\{ x \in G \mid xPx^{-1} = P\right\}$. Let $H$ be a subgroup of $G$, $H \supseteq N$. Prove: If $y \in G$ such that $yHy^{-1} = P$, then $y \in H$.

Let $G$ be a finite group. The probability of $G$ to be commutative is defined by

Let $p$ be a prime. Show that for any element $a \in \mathbb {Z}/p\mathbb {Z}$, there exist $b, c \in \mathbb {Z}/p\mathbb {Z}$ such that $a = b^2 + c^2$.

For a ring $R$, $R^*$ denotes its multiplicative group of units, $M_n(R)$ the ring of $n \times n$ matrices over $R$, and $GL_n(R) = M_n(R)^*$.

Describe an infinite set of integral solutions $(a, b)$ of the Pell's equation $x^2 - 2y^2 = 1$ (i.e., $a, b \in \mathbb {Z}$ such that $a^2 - 2b^2 = 1$).

Let $p$ be a prime. Let $R$ be a commutative ring in which $a^p = a$ for all $a \in R$.

Let $R$ be a commutative ring with $1$. If $R$ satisfies the a.c.c. (ascending chain condition) for finitely generated ideals then show that $R$ satisfies the a.c.c. for all ideals. Give an example (without proof) of such a ring which is an integral domain but not a p.i.d.

Let $R$ be a commutative ring. Let $A$ be an ideal of $R$.

Let $I$ be a nilpotent ideal in a ring $R$, let $M$ and $N$ be $R$-modules, and let $f : M \to N$ be an $R$-homomorphism. Show that if the induced map

is surjective, then $f$ is surjective.

Let $F$ be a finite field containing $5$ elements. Let $t$ be transcendental over $F$. Explicitly construct one non-archimedean absolute value $\left|\, \cdot \, \right|$ on $F(t)$. If $\overline{F(t)}$ is the completion of $F(t)$ with respect to $\left|\, \cdot \, \right|$, show that the set of $\alpha (t) \in \overline{F(t)}$ satisfying $\left|\alpha (t)\right| \leq 1$ is a local ring.

Prove that the ideal generated by $X_1, \ldots , X_n$ in the polynomial ring $\mathbb {C}^{X_1, \ldots , X_n}$ cannot be generated by fewer than $n$ elements.

Let $A$ be a commutative ring, and let $I$ and $J$ be two (proper) ideals such that every prime ideal of $A$ contains either $I$ or $J$ but no prime ideal contains both $I$ and $J$. Prove that

for some (nontrivial) rings $A_1$ and $A_2$.

Define $f : [0,1) \to [0,1)$ by

Find all $x$ such that

Let $T$ be the ring of all real trigonometric polynomials

Define $\deg f(x) = N$ where $a_N$ or $b_N \neq 0$. Show that $\deg f \cdot g = \deg f + \deg g$. Use this result to prove that $T$ is an integral domain which is not a unique factorization domain.

Let $A$ be a Noetherian integral domain integrally closed in its field of fractions $K$. Let $L$ be a finite separable field extension of $K$. If $B$ is the integral closure of $A$ in $L$, outline a proof that $B$ is Noetherian.

If $A$ is a commutative ring and $A[[x]]$ is the ring of formal power series over $A$, show that if $f = \sum_{n=0}^\infty a_n x^n$ is nilpotent, then all the elements $a_i \in A$ are nilpotent. If $A$ is Noetherian, prove the converse, i.e., that if all the elements $a_i$ are nilpotent, then $f$ is nilpotent.

Let $F$ be a field, and for $n \geq 1$ let $R_n$ be the subring of the ring of polynomials $F[x]$ consisting of polynomials

such that $f_1 = f_2 = \cdots = f_n = 0$.

Describe all subrings of $\mathbb {Q}$. (A subring contains $1$ by definition.)

Let $X$ be a finite set and $G$ a subgroup of the group of permutations of $X$. Define a relation $\sim$ on $X$ by requiring $x \sim y$ if either $x = y$ or the transposition $(x,y)$ (which interchanges $x, y \in X$ and leaves all other elements fixed) is an element of $G$. Show the following.

Let $p$ be an odd prime. Let $\zeta_p$ be a primitive $p$-th root of unity and $g$ a primitive root (mod $p$) (i.e., $g$ is a generator for $(\mathbb {Z}/p\mathbb {Z})^*$). Fix $e$, a divisor of $p-1$ and put $f = (p-1)/e$. Define

Show that, for any $i$, $\eta_i$ generates a subfield of $\mathbb {Q}(\zeta_p)$ of degree $e$ over $\mathbb {Q}$.

Let $K$ be a field and $x$ be an element of an extension of $K$ such that $x$ is transcendental over $K$. Put $G = \operatorname {Aut}(K(x)/K)$, and let $H$ denote the subgroup of $G$ consisting of the substitutions $x \to x + b$ with $b \in K$.

Let $E = \mathbb {C}(Y)$ with $Y$ an indeterminate, $F = \mathbb {C}(Z)$ with $Z = Y^n + Y^{-n}$, and $\zeta = e^{2\pi i/n}$.