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Determine, up to similarity, all $3\times 3$ matrices $𝐴$ over $\mathbb{Q}$ such that ${𝐴}^2+2{𝐴}^3+{𝐴}^4=0$ but $𝐴+{𝐴}^2\ne 0$.

Let $𝑅$ be a commutative ring with $1$, and suppose $𝐼$ and $𝐽$ are ideals of $𝑅$ such that $𝐼+𝐽=𝑅$. Show that:

(i) $𝐼𝐽=𝐼\cap 𝐽$,

(ii) $𝑅/(𝐼𝐽)\cong 𝑅/𝐼\oplus 𝑅/𝐽$.

Let $𝑝$ be a prime integer such that $𝑝\equiv 2\text{or}3(mod5)$. Prove that the polynomial

is irreducible over $\mathbb{Z}/𝑝\mathbb{Z}$.

Let $𝐺$ be a finite group and $𝑃$ a $𝑝$-Sylow subgroup. Let ${𝑁}_{𝐺(𝑃)}$ be the normalizer of $𝑃$ in $𝐺$. Show that:

(a) $𝑃$ is the unique $𝑝$-Sylow subgroup of ${𝑁}_{𝐺(𝑃)}$. (Do not quote a theorem that this is true!)

(b) ${𝑁}_{𝐺(𝑃)}$ is self-normalizing in $𝐺$.

Let ${𝑇}_1,{𝑇}_2,\dots ,{𝑇}_{𝑚}$ be linear operators on a vector space of dimension $𝑛$. Assume that

(i) $dimim({𝑇}_{𝑖})=1$, and

(ii) ${𝑇}_{𝑖}^2\ne 0$ and ${𝑇}_{𝑖}{𝑇}_{𝑗}=0$ for every $𝑖\ne 𝑗$.

Prove that $𝑚\le 𝑛$.

Let $𝑅$ be a commutative ring with $1$, and let $𝑆=𝑅[𝑥]$ be the polynomial ring in one variable. Let $𝑓\in 𝑆$. If $𝑓$ is a unit of $𝑆$ (that is, $𝑓$ is invertible in $𝑆$), show that $𝑓$ has the form $𝑓=𝑢+𝑔$, where $𝑢$ is a unit in $𝑅$ and $𝑔\in 𝑆$ is a nilpotent element without constant term.

Let $𝑘$ be a finite field of size $𝑞$.

(a) Prove that the number of $2\times 2$ matrices over $𝑘$ satisfying ${𝑇}^2=0$ is ${𝑞}^2$.

(b) Prove that the number of $3\times 3$ matrices over $𝑘$ satisfying ${𝑇}^3=0$ is ${𝑞}^6$.

Let $𝑉$ be a vector space over a field $𝐹$, and let $𝐵:𝑉\times 𝑉\to 𝐹$ be a symmetric bilinear form. This means that $𝐵$ is bilinear and $𝐵(𝑥,𝑦)=𝐵(𝑦,𝑥)$. Let $𝑞(𝑣)=𝐵(𝑣,𝑣)$.

(a) Show that if the characteristic of $𝐹$ is not 2 then $𝐵(𝑣,𝑤)=\frac{1}{2}(𝑞(𝑣+𝑤)-𝑞(𝑣)-𝑞(𝑤))$. (This obviously implies that if $𝑞=0$ then $𝐵=0$.)

(b) Give an example where the characteristic of $𝐹$ is 2 and $𝑞=0$ but $𝐵\ne 0$.

(c) Show that if the characteristic of $𝐹$ is not 2 or 3 and if $𝐵(𝑢,𝑣,𝑤)$ is a symmetric trilinear form, and if $𝑟(𝑣)=𝐵(𝑣,𝑣,𝑣)$, then $𝑟=0$ implies $𝐵=0$.

(a) Prove that if $𝐾$ is a field of finite degree over $\mathbb{Q}$ and ${𝑥}_1,\dots ,{𝑥}_{𝑛}$ are finitely many elements of $𝐾$ then the subring $\mathbb{Z}[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ they generate over $\mathbb{Z}$ is not equal to $𝐾$. (Hint: Show they all lie in ${\mathrm{𝒪︀}}_{𝐾}[1/𝑎]$ for a suitable nonzero $𝑎$ in ${\mathrm{𝒪︀}}_{𝐾}$, where ${\mathrm{𝒪︀}}_{𝐾}$ denotes the integral closure of $\mathbb{Z}$ in $𝐾$.)

(b) Let $𝔪$ be a maximal ideal of $\mathbb{Z}[{𝑥}_1,\dots {𝑥}_{𝑛}]$ and $𝐹=\mathbb{Z}[{𝑥}_1,\dots ,{𝑥}_{𝑛}]/𝔪$. Use (a) and the Nullstellensatz to show that $𝐹$ cannot have characteristic 0, and then deduce for $𝑝=char(𝐹)$ that $𝐹$ is of finite degree over ${𝔽}_{𝑝}$ (so $𝐹$ is actually finite).

Let $𝐺$ be a finite group.

(a) Let $𝜋:𝐺\to GL(𝑉)$ be an irreducible complex representation, and let $𝜒$ be its character. If $𝑔\in 𝐺$, show that $|𝜒(𝑔)|=dim(𝑉)$ if and only if there is a scalar $𝑐\in ℂ$ such that $𝜋(𝑔)𝑣=𝑐𝑣$ for all $𝑣\in 𝑉$.

(b) Show that $𝑔$ is in the center $𝑍(𝐺)$ if and only if $|𝜒(𝑔)|=𝜒(1)$ for every irreducible character $𝜒$ of $𝐺$.