# Algebra Exercises

#### Abstract Algebra

• Let $F$ be a field of characteristic $\neq 2$. Let $a,b$ be elements of the field $F$ with $b$ not a square in $F$. Prove that a necessary and sufficient condition for $\sqrt{a+\sqrt{b}} = \sqrt{m}+\sqrt{n}$ for some $m$ and $n$ in $F$ is that $a^{2}-b$ is a square in $F$. Use this to determine when the field $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ ($a,b \in \mathbb{Q}$) is biquadratic over $\mathbb{Q}$.
• Suppose $\alpha$ is a rational root of a monic polynomial in $\mathbb{Z}[x]$. Prove that $\alpha$ is an integer.
• Let $V$ be a vector space of finite dimension. If $\varphi$ is a linear transformation from $V$ to $V$ prove there is an integer $m$ such that the intersection of the image of $\varphi^{m}$ and the kernel of $\varphi^{m}$ is ${0}$.
• Show that $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q}$ and $\mathbb{Q} \otimes_{\mathbb{Q}} \mathbb{Q}$ are isomorphic left $\mathbb{Q}$-modules.
• Show that if $M_{1}$ and $M_{2}$ are irreducible $R$-modules, then any nonzero $R$-module homomorphism from $M_{1}$ to $M_{2}$ is an isomorphism. Deduce that if $M$ is irreducible then $\text{End}_{R}(M)$ is a division ring (this result is called Schur’s Lemma).
• If $M$ is a finite abelian group then $M$ is naturally a $\mathbb{Z}$-module. Can this action be extended to make $M$ into a $\mathbb{Q}$-module? Prove or provide a counterexample.
• Prove that $\mathbb{R}[x]/(x^{2}+1)$ is a field which is isomorphic to the complex numbers.
• Prove that the ring $M_{2}(\mathbb{R})$ contains a subring that is isomorphic to $\mathbb{C}$.
• Let $F_{1}$ and $F_{2}$ be free groups of finite rank. Prove that $F_{1} \cong F_{2}$ if and only if they have the same rank. What facts do you need in order to extend your proof to infinite ranks (where the result is also true)?
• Prove that a finite abelian group is the direct product of its Sylow subgroups.
• For any group $G$ define the dual group of $G$ (denoted $\hat{G}$) to be the set of all homomorphisms from $G$ into the multiplicative group of roots of unity of $\mathbb{C}$. Define a group operation in $\hat{G}$ by pointwise multiplication of functions. Show that this operation on $\hat{G}$ makes $\hat{G}$ into an abelian group. Moreover, show that if $G$ is a finite abelian group, prove that $\hat{G} \cong G$.
• Prove that the group of rigid motions in $\mathbb{R}^{3}$ of an icosahedron is isomorphic to $A_{5}$. Prove that the analogous group for the dodecahedron is isomorphic to $A_{5}$ as well.
• Let $p$ be a prime and let $G$ be the group of $p$-power roots of $1$ in $\mathbb{C}$. Prove that the map $z \mapsto z^{p}$ is a surjective homomorphism. Deduce that $G$ is isomorphic to a proper quotient of itself.
• Prove that $\mathbb{Q} \times \mathbb{Q}$ is not cyclic.
• Let $G$ be a group, and consider the bijective map $g \mapsto g^{-1}$. For which $G$ is this map an automorphism?
• Let $\mathbb{F}_{q}$ be a finite fields of prime characteristic $p$ with $q$ elements. Find the order of $GL_{n}(\mathbb{F}_{q})$, the highest power of $p$ dividing the order of $GL_{n}(\mathbb{F}_{q})$, and show that the subgroup of $GL_{n}(\mathbb{F}_{q})$ consisting of upper triangular matrices with all diagonal entries $1$ is a $p$-Sylow subgroup.
• Let $p$ be a prime. Find all abelian groups of order $p$.

#### Linear Algebra

• Prove or give a coutner example: if $U_{1}$, $U_{2}$, and $W$ are subspaces of a vector space $V$ such that $V = U_{1} \oplus W$ and $V = U_{2} \oplus W$, then $U_{1} = U_{2}$.
• Prove that if $U_{1},\ldots,U_{m}$ are subspaces of $V$ such that $V = U_{1} \oplus \cdots \oplus U_{m}$, then $dim(V) = dim(U_{1})+\cdots+dim(U_{m})$.
• Let $V$ be a finite dimensional vector space and $U$ be a subspace of $V$. Show that any linear map $T$ on $U$ can be extended to a linear map on $V$.
• Prove that if $T$ is a linear map on a vector space $V$ will null space and range both finite dimensional, then $V$ is finite dimensional.
• Let $T$ be a linear operator on $\mathbb{C}^{2}$ defined by $T(w,z) = (z,w)$. Find all eigenvalues and eigenvectors of $T$.
• Suppose $T$ is a linear operator and the dimension of the rang of $T$ is $k$. Prove that $T$ has at most $k+1$ distinct eigenvalues.
• Let $S$ and $T$ be linear operators on a finite dimensional complex vector space such that $ST = TS$. Prove $S$ and $T$ have a common eigenvector.
• Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagional, but the operator is not invertible.
• Let $P$ be a projection map on a vector space $V$ (i.e., $P^{2} = P$). Prove $V$ is the direct sum of the null space and range of $P$.