Algebra Qual 2024-01

Problem 1

Let $G$ be a group, $H \leq G$ a subgroup that is not normal. Prove there exist cosets $H a$ and $H b$ such that $H a H b \neq H a b$ .

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Let $G$ be a group, $H\leq G$ a subgroup that is not normal. Prove there exist cosets $Ha$ and $Hb$ such that $HaHb\neq Hab$.

Problem 2

Determine with proof the automorphism group $Aut (V)$ of the Klein 4-group $V = {e, a, b, a b}$ . To what familiar group is it isomorphic?

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Determine with proof the automorphism group $\operatorname{Aut}(V)$ of the Klein 4-group $V=\{e,a,b,ab\}$. To what familiar group is it isomorphic?

Problem 3

Suppose $R$ is a finite ring with no nontrivial zero-divisors. Prove that $R$ contains an element $1$ satisfying $1 \cdot a = a \cdot 1 = a$ for all $a \in R$ .

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Suppose $R$ is a finite ring with no nontrivial zero-divisors. Prove that $R$ contains an element $1$ satisfying $1\cdot a=a\cdot 1=a$ for all $a\in R$.

Problem 4

Let $k \subset K$ be fields, and let $k [X]$ be the polynomial ring in one variable with coefficients in $k$ . The evaluation at $z \in K$ is a ring homomorphism $ε : k [X] \to K$ defined by $ε (f (X)) = f (z)$ . Prove that if $ε$ is not injective, then $ε (k [X])$ is a field.

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Let $k\subset K$ be fields, and let $k[X]$ be the polynomial ring in one variable with coefficients in $k$. The {\bfseries evaluation} at $z\in K$  is a ring homomorphism $\varepsilon:k[X]\to K$ defined by $\varepsilon(f(X))=f(z)$. Prove that if $\varepsilon$ is not injective, then $\varepsilon(k[X])$ is a field.

Problem 5

Let $V$ be a vector space with basis $v_{0}, \dots, v_{n}$ and let $a_{0}, \dots, a_{n}$ be scalars. Define a linear transformation $T : V \to V$ by the rules $T (v_{i}) = v_{i + 1}$ if $i < n$ , and $T (v_{n}) = a_{0} v_{0} + a_{1} v_{1} + \dots + a_{n} v_{n}$ . You don't have to prove this defines a linear transformation. Determine the matrix for $T$ with respect to the basis $v_{0}, \dots, v_{n}$ , and determine the characteristic polynomial of $T$ .

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Let $V$ be a vector space with basis ${\bf v}_0,\ldots, {\bf v}_n$ and let $a_0,\ldots, a_n$ be scalars. Define a linear transformation $T:V\to V$ by the rules $T({\bf v}_i)={\bf v}_{i+1}$ if $i<n$, and $T({\bf v}_n)=a_0{\bf v}_0+a_1{\bf v}_1+\cdots +a_n {\bf v}_n$. You don't have to prove this defines a linear transformation. Determine the matrix for $T$ with respect to the basis ${\bf v}_0,\ldots, {\bf v}_n$, and determine the characteristic polynomial of $T$.