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A map of commutative rings $𝑓:𝐴\to 𝐵$ is faithfully flat if it is flat and $𝑁{\otimes }_{𝐴}𝐵\ne 0$ for all nonzero $𝐴$-modules $𝑁$. (There are other equivalent definitions, but use this one for the questions below.)

(a) Prove that if $𝐴$ and $𝐵$ are local rings and $𝑓$ is a flat map that is local (i.e., ${𝑓}^{-1}({𝔪}_{𝐵})={𝔪}_{𝐴}$) then $𝑓$ is faithfully flat.

(b) Let $Spec(𝑅)$ be the set of prime ideals of a commutative ring $𝑅$. Prove that $𝑓$ is faithfully flat if and only if $𝑓$ is flat and the map $Spec(𝑓):Spec(𝐵)\to Spec(𝐴)$ sending $𝔭$ to ${𝑓}^{-1}(𝔭)$ is surjective. (Hint: for $\Leftarrow$, apply (a) to a suitable localization.)

(c) Assume $𝑓$ is faithfully flat. For submodules $𝑀,{𝑀}^{\prime }$ of $𝑁$ prove $𝑀\subset {𝑀}^{\prime }$ inside $𝑁$ if $𝑀{\otimes }_{𝐴}𝐵\subset {𝑀}^{\prime }{\otimes }_{𝐴}𝐵$ inside $𝑁{\otimes }_{𝐴}𝐵$. (Hint: Consider $𝑀+{𝑀}^{\prime }\subset 𝑁$.) Deduce that if $𝑎,{𝑎}^{\prime }\in 𝐴$ satisfy $𝑓(𝑎)|𝑓({𝑎}^{\prime })$ in $𝐵$ then $𝑎|{𝑎}^{\prime }$ in $𝐴$.

A Dedekind domain is a commutative ring that is a Noetherian integrally closed integral domain that is not a field and in which every nonzero prime ideal is maximal. For example, a PID that is not a field is a Dedekind domain.

(a) If $𝐴$ is a Dedekind domain and $𝐼,𝐽$ are nonzero ideals then prove that the natural surjective map $𝐼{\otimes }_{𝐴}𝐽\to 𝐼𝐽$ defined by $𝑎\otimes 𝑏\mapsto 𝑎𝑏$ is an isomorphism. (You may use without proof the fact that every local Dedekind domain is a discrete valuation ring.)

(b) Prove that the conclusion of (a) is false $𝐴=ℂ[𝑥,𝑦]$ and each $𝐼,𝐽$ is equal to the maximal ideal $𝔪=(𝑥,𝑦)$.

The discriminant of a polynomial $𝑓(𝑋)={\prod }_{𝑖=1}^{𝑛}(𝑋-{𝑟}_{𝑖})$ is

Let $𝑓(𝑋)\in \mathbb{Q}[𝑋]$ be a monic irreducible polynomial of degree 4 with roots $𝛼,𝛽,𝛾,𝛿$.

(a) Prove that $𝛼𝛽+𝛾𝛿$, $𝛼𝛾+𝛽𝛿$, and $𝛼𝛿+𝛽𝛾$ are the roots of a monic cubic polynomial $𝑔(𝑋)\in \mathbb{Q}[𝑋]$ whose discriminant is the same as the discriminant of $𝑓$.

(b) If $𝑓\in \mathbb{Z}[𝑋]$ prove that $𝑔\in \mathbb{Z}[𝑋]$.

(c) Explain briefly why the Galois group of $𝑓$ over $\mathbb{Q}$ is one of the five groups ${𝑆}_4,{𝐴}_4,{𝑍}_4,{𝐷}_4$, or ${𝑍}_2\times {𝑍}_2$ (where ${𝑍}_{𝑛}$ is the cyclic group of order $𝑛$). In which cases is $𝑔$ an irreducible polynomial?

(a) Let $𝐴\subseteq 𝐵$ be commutative rings. Define what it means for $𝐵$ to be integral over $𝐴$. Prove that if $𝐵$ is finitely generated as an $𝐴$-module then it is integral over $𝐴$.

(b) Prove that if $𝐴\subseteq 𝐵$ are domains with $𝐵$ integral over its subring $𝐴$ then $𝐴$ is a field if and only if $𝐵$ is a field.

(c) Let $𝐴$ be a domain with field of fractions $𝐾$. Let $𝐿$ a finite extension of $𝐾$. Let $𝐵$ be the set of elements of $𝐿$ which are integral over $𝐴$. Prove that $𝐵$ is a subring of $𝐿$ and that $𝐿$ is its field of fractions.

Let $𝑘$ be an algebraically closed field.

(a) State the Nullstellensatz for ideals in $𝑘[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$, and explain why a pair of radical ideals $𝐽,{𝐽}^{\prime }\subset 𝑘[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ with associated zero loci $𝑍,{𝑍}^{\prime }\subset {𝑘}^{𝑛}$ satisfy $𝐽\supset {𝐽}^{\prime }$ if and only if $𝑍\subset {𝑍}^{\prime }$. Use this to describe the maximal ideals in $𝑘[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$.

(b) Define the Zariski topology on ${𝑘}^{𝑛}$ and check it is a topology.

(c) Let $𝐽\subset 𝑘[{𝑥}_1,\dots ,{𝑥}_{𝑛}]$ be a radical proper ideal. Prove that the corresponding non-empty zero locus $𝑍\subset {𝑘}^{𝑛}$ is irreducible for the Zariski topology (i.e., not a union of two proper closed subsets) if and only if $𝐽$ is a prime ideal.

Let $𝑉$ be the $ℝ$-vector space ${ℝ}^{2𝑛}$ with the bilinear form $𝐵:𝑉\times 𝑉\to ℝ$ given by

The form $𝐵$ is nondegenerate; that is, $𝐵(𝑥,𝑉)=0$ implies $𝑥=0$. It is alternating; that is, $𝐵(𝑥,𝑥)=0$. Since the characteristic is not 2, this is equivalent to saying that it is skew-symmetric; that is, $𝐵(𝑥,𝑦)=-𝐵(𝑦,𝑥)$. If $𝑈$ is a vector subspace of $𝑉$, let ${𝑈}^{⟂}$ be the orthogonal complement; that is, ${𝑈}^{⟂}=\{𝑥\in 𝑉|𝐵(𝑥,𝑈)=0\}$.

(a) Explain briefly why $dim(𝑈)+dim({𝑈}^{⟂})=2𝑛$.

(b) A subspace $𝑊$ of $𝑉$ is called isotropic if $𝐵(𝑊,𝑊)=0$; that is, if $𝑊\subseteq {𝑊}^{⟂}$. Show that if $𝑊$ is isotropic of dimension $𝑛$ then there exists another isotropic subspace ${𝑊}^{\prime }$ such that $𝑉=𝑊\oplus {𝑊}^{\prime }$.

(c) Let $𝐺$ be the group of $𝑔\in GL(𝑉)$ such that $𝐵(𝑔𝑥,𝑔𝑦)=𝐵(𝑥,𝑦)$ for all $𝑥,𝑦\in 𝑉$. Show that $𝐺$ is transitive on the set of isotropic subspaces of dimension $𝑛$.

For each action of ${𝑆}_4$, identify the indicated stabilizer.

(a) The natural action on $\{1,2,3,4\}$; stabilize $2$.

(b) Left multiplication on ${𝑆}_4$; stabilize $(23)$.

(c) Conjugation on ${𝑆}_4$; stabilize $(23)$.

Let $𝑝$ be prime, let $𝐾$ be the splitting field over $\mathbb{Q}$ of an irreducible polynomial $𝑓$ of degree $𝑝$, and let $𝛼,𝛽$ be distinct roots such that

Prove that $Gal(𝐾/\mathbb{Q})\cong \mathbb{Z}/𝑝\mathbb{Z}$.

Let $𝐺$ be a group whose automorphism group is cyclic. Prove that $𝐺$ is abelian.

Let $𝑅$ be a nonzero finite commutative ring with identity. Prove that every prime ideal of $𝑅$ is maximal.