Ext Functors and Module Extensions

What about the Homomorphisms in Problem 4? Namely,

1. Is it true that:

   \[ \operatorname {Hom}_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=k\cdot \operatorname {id} \]

   is a one-dimensional vector space, or are there more \(k[x, y]\)-module homomorphisms than just the multiples of the identity?

2. Are there any non-split extensions of \(k[x, y]\)-modules:

   \[ 0\to \langle x,y\rangle \to M\to \langle x,y\rangle \to 0 \]

   i.e. is

   \(\operatorname {Ext}^1_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=0\) or not?

Suppose \(R=\mathbb {Q}[x,y]\) is a polynomial ring. Compute the \(\mathbb {Q}\)-vector space dimension of \[ \operatorname {Tor}^R_1\left (\operatorname {Ext}^1_R(R/(x^2),R/(y^2)),R/(y)\right ). \]

Let \(R=(\mathbb {Z}/3\mathbb {Z})[x,y]\) and let \(I=(x,2y)\). Compute the number of elements in \(\operatorname {Ext}^i(R/I,R)\) for each \(i\ge 0\).

Let \(2^{1/4}\) be the unique positive real fourth root of of 2 in \(\mathbb {R}\). Show that \(\mathbb {Q}(2^{1/4})/\mathbb {Q}\) is not a Galois extension.

Let \(R:=\mathbb {Z}[x]\) and \(I\) the ideal \((3,x)\). How many elements are in \(\operatorname {Ext}^i(R/I,R)\) for \(i=0,1,2\)?

With \(\alpha =\sqrt {6+\sqrt {11}}\), prove that the extension \(\mathbb {Q}[\alpha ]/\mathbb {Q}\) is Galois and compute its Galois group.

Describe a primitive generator for the degree two extension of \(\mathbb {Q}\) inside the extension \(\mathbb {Q}(\zeta )\), where \(\zeta :=e^{2\pi i/7}\). Justify that the extension you identify is of degree two.

Suppose that \(R\) is a commutative ring with unity, and \(N\subseteq M\) are \(R\)-modules. If \(\operatorname {Ext}^1(M/N,N)=0\), prove that the inclusion \(N\hookrightarrow M\) is split.

Determine the extension degree of the splitting field of \(x^7-1\) over \(\mathbb {F}_{11}\).

Let \(M/K\) be a Galois extension of degree 700. Prove that there exists an intermediate field \(L\) with \([L:K]=28\). Is \(L/K\) necessarily Galois?

Find a Galois extension of \(\mathbb {Q}\) with Galois group isomorphic to \(\mathbb {Z}/3\times \mathbb {Z}/3\).

Let \(F\subset L\) a extension of fields of degree 4. Prove that there are no more than 3 fields proper intermediate subfields \(K\); namely, such that \(F\subset K\subset L\).

Let \(F\) be a field and \(n\) a positive integer such that \(F\) has no nontrivial field extensions of degree less than \(n\). Let \(L=F[\alpha ]\) be an extension such that \(\alpha ^n\) is in \(F\). Prove that each element of \(L\) is a product of elements of the form \(a\alpha +b\), with \(a,b\) in \(F\).

Let \(\alpha =2^{1/5}+5^{1/2}\). Determine the extension degree of \(\alpha \) over \(\mathbb {Q}\).

Let \(n\) a positive integer, and let \(\mathbf {x}:=x_1,\ldots ,x_n\) be indeterminates over \(\mathbb {Q}\). For each positive integer \(i\) set \[ \alpha _i=x_1^i+\cdots +x_n^i. \]

1. Clearly \(\mathbb {Q}(\alpha _1,\ldots ,\alpha _n)\subseteq \mathbb {Q}(s_1,\ldots ,s_n)\), the field of symmetric functions in the \(\mathbf {x}\). What is the degree of this extension?
2. Prove the characteristic polynomial of an \(n\times n\) matrix \(A\) over \(\mathbb {Q}\) is completely determined by the elements \(\operatorname {trace}(A),\ldots ,\operatorname {trace}(A^n)\).

Let \(a,b,c,d\) integers such that \[ \det (A)\ne 0\qquad \text {where}\qquad A=\begin {pmatrix} a & b\\ c & d \end {pmatrix}. \] Let \(x\) and \(y\) be indeterminates, and set \(L=\mathbb {C}(x,y)\) and \(K=\mathbb {C}(x^a y^b,x^c y^d)\).

1. Prove that \(L\) is a finite extension of \(K\) and that \((L:K)=|\det (A)|\).
2. Show that \(L/K\) is Galois, and compute the Galois group.

Let \(R=\mathbb {Q}[x,y]\) and let \(I=\langle x,y\rangle \). Compute the vector space rank of \(\operatorname {Ext}^{1}_{R}(I,R)\) over \(\mathbb {Q}\).

Let \(E\) be the splitting field of \(f(x)=x^{14}+1\) over \(\mathbb {F}_2\), and let \(K\) be the splitting field \(g(x)=x^{21}+1\) over \(\mathbb {F}_2\). Prove that \(K\) contains an isomorphic copy of \(E\), and compute the extension degree of \(K\) over \(E\).

Let \(\mathbb {F}\) be a field and let \(A\) be the polynomial ring \(\mathbb {F}[x_1,x_2,\ldots ,x_n]\). Regard \(\mathbb {F}\) as an \(A\)-module via the map \(A\longrightarrow A/(x_1,x_2,\ldots ,x_n)A\). Prove that \[ \operatorname {Tor}^{A}_{i}(\mathbb {F},\mathbb {F})\cong \operatorname {Ext}^{i}_{A}(\mathbb {F},\mathbb {F})\cong \bigwedge ^{i}(\mathbb {F}^{n}) \quad \text {for each } i\geq 0. \]

Let \(\mathbb {F}\) be a field with \(\mathbb {Q}\subset \mathbb {F}\subset \mathbb {C}\) such that \([\mathbb {F}:\mathbb {Q}]\) is odd.

1. If \(\mathbb {F}/\mathbb {Q}\) is Galois, prove that \(\mathbb {F}\) is contained in \(\mathbb {R}\).
2. Find an extension with \([\mathbb {F}:\mathbb {Q}]=3\) such that \(\mathbb {F}\) is not contained in \(\mathbb {R}\).

Let \(F\) be a field that contains a primitive \(n\)th root of unity. Show that if \(a\) is an element of \(F\) and the field \(E\) is obtained from \(F\) by adjoining an \(n\)th root of \(a\), then \(E\) is a Galois extension of \(F\) with cyclic Galois group.

Show that there exists a Galois extension of the field \(\mathbb {Q}\) of rational numbers with Galois group \(\mathbb {Z}/5\mathbb {Z}\).

Find a real number \(\alpha \) such that \(\mathbb {Q}(\alpha )\) is a Galois extension of \(\mathbb {Q}\) with the Galois group \(\mathbb {Z}/5\mathbb {Z}\).

Let \(\zeta _8=e^{\frac {\pi }{4}i}\). Determine the Galois group of \(K=\mathbb {Q}(\zeta _8)\) and then determine all quadratic extensions of \(\mathbb {Q}\) contained in \(K\).

Let \(G\) be a finite group, let \(H\) be a subgroup of \(G\), and let \(\mathbb {C}[G]\) (respectively, \(\mathbb {C}[H]\)) denote the complex group algebra of \(G\) (respectively, \(H\)). Let \(I_H^G\) denote the functor from the category of finite-dimensional \(\mathbb {C}[H]\) modules to the category of finite-dimensional \(\mathbb {C}[G]\) modules which, on the level of objects, is defined by \[ I_H^G(M)=\mathbb {C}[G]\otimes _{\mathbb {C}[H]}M. \] Prove or disprove: \(I_H^G\) is an exact functor.

Let \(G\) be a finite group and let \(\mathbb {C}[G]\) denote the complex group algebra of \(G\). Show that the invariants functor which (on the level of objects) takes a \(\mathbb {C}[G]\) module \(M\) to \[ M^G:=\{m\in M\mid f\cdot m=m\text { for all }f\in \mathbb {C}[G]\} \] is an exact functor.

Let \(K\) and \(F\) be fields such that \(K\) is a finite extension of \(F\). Assume that there are only finitely many subfields \(L\) such that \(F\subset L\subset K\). Show that \(K\) is a primitive extension of \(F\) (i.e. \(K=F(\theta )\) for some \(\theta \in K\)).

Find a galois extension \(K\) of \(\mathbb {Q}\) with Galois group \(\mathbb {Z}/5\mathbb {Z}\oplus \mathbb {Z}/5\mathbb {Z}\).

Let \(R\) be a commutative ring with identity. Given an \(R\)-module \(M\), recall the \(R\)-module \[ \bigwedge M=\bigoplus _{j\ge 0}\bigwedge ^j M, \] where \(\bigwedge ^j M\) is the \(j\)th exterior power of \(M\). Define a natural operation on morphisms so that the assignment \(M\mapsto \bigwedge M\) becomes a functor and determine whether this functor is exact.

Let \(E\) be an extension of the field \(F\) and let \(\alpha \) be an element of \(E\) which is algebraic over \(F\) of degree \(n\).

1. If \(n\) is prime to 2, prove that \(F(\alpha ^2)=F(\alpha )\).
2. Give an example in which \(n\) is prime to 3 but \(F(\alpha ^3)\ne F(\alpha )\).

Let \(E\subset F\) be an extension of finite fields. Prove that the trace map from \(F\) to \(E\) is surjective.

Let \(G\) be a finite group, and let \(C\) be the category of finitely generated \(\mathbb {Q}[G]\)-modules. Show the the functor which takes a \(\mathbb {Q}[G]\)-module to its submodule of invariants (that is, the set of \(m\in M\) such that \(gm=m\) for all \(g\in G\)) is exact.

Show that there exists a Galois extension of the rational numbers with Galois group \(\mathbb {Z}/7\mathbb {Z}\).

Let \(F\) be a field, and let \(f(X)\) be a polynomial in \(F[X]\). Show that \(f(X)\) has a multiple root in some extension of \(F\) if and only if \(f(X)\) and its derivative \(f'(X)\) are not relatively prime.

Let \(E\) be a normal extension of \(\mathbb {Q}\) contained in \(\overline {\mathbb {Q}}\) (the algebraic closure of \(\mathbb {Q}\)), and let \(K\) be an extension of \(\mathbb {Q}\) contained in \(E\). Let \(K'\) be a subfield of \(\overline {\mathbb {Q}}\) which is isomorphic to \(K\). Show that \(K'\) is contained in \(E\) and the Galois groups of \(E/K\) and \(E/K'\) are conjugate in the Galois group of \(E/\mathbb {Q}\).

Let \(K/k\) be a Galois extension with Galois group \(G\). Show that every 1-cycle on \(G\) with values in \(K^*\) is a 1-coboundary.

Let \(E\) be an extension of the field \(F\) and let \(\alpha \) be an element of \(E\) which is algebraic over \(F\) of degree \(n\).

1. If \(n\) is prime to 2, prove that \(F(\alpha ^2)=F(\alpha )\).
2. Give an example in which \(n\) is prime to 3 but \(F(\alpha ^3)\ne F(\alpha )\).

Let \(E\) be a finite extension of the rationals. Show that the number of roots of unity in \(E\) is finite.

Suppose \(F\) is a field of characteristic 0 with the property that every polynomial of odd degree over \(F\) has at least one root in \(F\). Assume that \(E\) is an extension of degree 2 over \(F\) with the property that for each \(\alpha \in E\), the polynomial \(x^2-\alpha \) has a root in \(E\). Prove that \(E\) is algebraically closed.

Let \(F\) be the field of integers modulo 7. Determine which finite extensions of \(F\) contain exactly four 4th roots of 1.

1. Explain what it means for a contravariant functor \(F\colon \mathcal {C}\to \mathbf {Set}\) to be representable (‘Set’ is the category of sets).
2. Show how the power set \(P(S)=\{T\mid T\subseteq S\}\) defines a contravariant functor \(P\colon \mathbf {Set}\to \mathbf {Set}\) and prove that \(P\) is representable.