Polynomial Rings and Ring Homomorphism Kernels

Suppose that \(A\) is an \(n \times n\) matrix with characteristic polynomial

Prove that if \(A^d=A^e\) for any pair \(d\ne e\) of integers, then \(A=I_n\). And no, you cannot just quote the theorem that a matrix of finite order is diagonalizable. We want you to demonstrate that you know why it is true.

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?

Identify all the ideals in the ring \(R=\mathbb {Z}[x]/(6,x^3)\).

Determine all the possible Jordan canonical forms for a linear transformation with characteristic polynomial

Prove the following assertions for the ring \(S:=\mathbb {R}[x]/(x^2+1)^2\).

1. There exist exactly two ring homomorphisms \(\pi :S\to \mathbb {C}\) such that \(\pi |_{\mathbb {R}}=\operatorname {id}_{\mathbb {R}}\), the identity map on \(\mathbb {R}\).
2. Choose one of the homomorphisms in (a), and call it \(\pi \). Prove that there exists a unique homomorphism of rings \(\sigma :\mathbb {C}\to S\) such that \(\sigma |_{\mathbb {R}}=\operatorname {id}_{\mathbb {R}}\) and \(\pi \sigma =\operatorname {id}_{\mathbb {C}}\).
3. Prove that \(\sigma \) extends to an isomorphism of rings \(\mathbb {C}[y]/(y^2)\to S\).

Suppose \(R\) is a commutative ring with 1 and \(x, y \in R\) satisfy \(xy = 0\) and \((x, y) = R\) (i.e. the ideal generated by \(x\) and \(y\) is the whole ring). Prove that the \(R\)-module homomorphism

sending \(f\in R\) to its image in \(R/(x)\), and in \(R/(y)\), is an isomorphism.

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

Determine up to similarity, all real \(5\times 5\)-matrices with characteristic polynomial \(x(x^2+1)^2\).

Find every prime ideal \(Q\) of \(R=\mathbb {Z}[x]/(x^3-3x,4+x)\) and count the number of elements in each quotient ring

\(R/Q\).

Suppose that \(R\) is a commutative ring and \(0\to M\xrightarrow {\alpha }N\xrightarrow {\beta }P\to 0\) is a short exact sequence of \(R\)-modules and \(L\) is an \(R\)-module. Prove, via an explicit argument involving module homomorphisms (and without citing properties of the \(\operatorname {Hom}\) functor), that there is an exact sequence of Abelian groups: \[ 0\to \operatorname {Hom}_R(P,L)\xrightarrow {\bar {\beta }}\operatorname {Hom}_R(N,L)\xrightarrow {\bar {\alpha }}\operatorname {Hom}_R(M,L) \]

You must concretely explain how to induce the maps \(\bar {\alpha }\) and \(\bar {\beta }\) from \(\alpha \) and \(\beta \) respectively.

Consider the ideal \(I:=(2x-9,3x-7)\) in the ring \(\mathbb {Z}[x]\). Find the smallest positive integer \(n\) such that

belongs to the ideal \(I\).

Let \(R:=\mathbb {Q}[x]/(x^3-1)\). Give an example of a finitely generated projective \(R\)-module that is not free.

Consider the ring homomorphism given by

Determine a minimal generating set for the kernel.

Let \(R:=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^3\xrightarrow {\begin {pmatrix}x-2&4&x\\0&4&x\\0&-x&-1\end {pmatrix}}R^3. \]

Write \(M\) as a direct sum of cyclic \(R\)-modules.

Consider the ring homomorphism given by

Is this homomorphism surjective? Determine a minimal generating set for the kernel.

Let \(R=\mathbb {Z}[x]\) and suppose that \(I=\langle x\rangle \) and that \(J=\langle x,5\rangle \). How many elements are in \(\operatorname {Tor}^R_i(R/I,R/J)\) for \(i=0,1,2\)?

Find a polynomial \(f(x)\in \mathbb {F}_2[x]\) such that \(\mathbb {F}_2[x]/f(x)\) is a field with 8 elements.

Let \(R\) be a commutative Noetherian ring, and let \(\varphi :R\longrightarrow R\) be a surjective ring homomorphism. Is \(\varphi \) necessarily an isomorphism?

Suppose \(M\) is a \(3\times 3\) matrix, with entries from the field of real numbers, such that \[ M^8=I\qquad \text {and}\qquad M^4\ne I. \] Determine the possibilities for the minimal polynomial of \(M\).

Let \(I\) be the ideal \((x,y)\) in the polynomial ring \(R:=\mathbb {C}[x,y]\), and let \(\Lambda ^2(I)\) denote the second exterior power of \(I\) in the category of \(R\)-modules. Prove or disprove: \(\Lambda ^2(I)\) is zero.

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

Consider \(R:=\mathbb {Q}[x]\) and the matrix: \[ A= \begin {bmatrix} x-1 & 0\\ 1-x & x^2 \end {bmatrix}. \] Let \(M\) be the cokernel of the map \(R^2\longrightarrow R^2\) given by multiplication by \(A\). Compute the rank of the \(\mathbb {Q}\)-vector space \(\operatorname {Hom}_R(M,R/(x^2))\).

Describe all the prime ideals in the ring \(\mathbb {Z}[x]/(x^3+1,6)\).

For each positive integer \(n\) let \(R_n=\mathbb {Z}[2^{1/2},\ldots ,n^{1/n}]\) viewed as a subring of the field of real numbers. Prove that the ring \(R:=\bigcup _{n\ge 1}R_n\) is not Noetherian.

Suppose \(\varphi :G\longrightarrow H\) is a group homomorphism and \(H\) is a group of order 33. Further suppose that \(N\supseteq \ker \varphi \) is a subgroup of \(G\). Prove that \(N\) is normal.

Suppose \(k\) is a field and suppose that \(f(x)\in k[x]\) has degree \(n\). Prove that \(f(x)\) has no repeated factors in \(k[x]\) if and only if all \(n\times n\) matrices over \(k\) with characteristic polynomial \(f(x)\) are similar.

Let \(S_n\) denote the symmetric group on \(n\) elements. Does there exist an injective homomorphism \(S_6\longrightarrow S_5\times S_5\)?

Let \(R=\mathbb {Q}[x]\) and consider the submodule \(M\) of \(R^2\) generated by the elements \((x^2-1,x-1)\) and \((x^2+x,x)\). Write \(M\) as a direct sum of cyclic modules.

Let \(K\) be a field, and let \(L\) be the splitting field of an irreducible and separable polynomial \(g(x)\in K[x]\). If \(\operatorname {Gal}(L/K)\) is abelian, and \(\alpha \in L\) is a root of \(g(x)\), prove that \(L=K(\alpha )\).

Let \(A=\mathbb {F}_3[x]\), i.e., \(A\) is a polynomial ring in one variable over the field with 3 elements. Suppose \(M\) and \(N\) are finitely generated \(A\)-modules such that \[ M\oplus \frac {A}{x^3+1} \cong N\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}. \] Are \(M\) and \(N\) isomorphic?

Determine the possible characteristic and minimal polynomials of an \(n\times n\) matrix over \(\mathbb {C}\) that has rank 1.

Find all the prime ideals of \(\mathbb {Z}[x]/(30,x^3+1)\) and identify which are maximal.

Find all prime ideals of the ring \(\mathbb {Z}[x]/(15,x^3-2)\).

Prove that if a \(3\times 3\) matrix \(A\) over \(\mathbb {Q}\) satisfies \(A^8=I\), then \(A^4=I\). Justify claims of irreducibility of polynomials.

Compute the Galois group of the polynomial \(x^{10}+x^5+1\) over \(\mathbb {Q}\).

Let \(R=\mathbb {Z}[i]\) and \(S=\mathbb {Z}[\sqrt {-3}]\). Does there exist a ring homomorphism \(R\longrightarrow S\)? Does there exist a ring homomorphism \(S\longrightarrow R\)?

Determine, up to conjugacy, all real \(5\times 5\) matrices with characteristic polynomial \(x(x^2+1)^2\).

Determine the minimal polynomial of \(\pi \) over the field \(\mathbb {Q}((\pi ^2+1)/(\pi -1))\).

(Use, without proof, that \(\pi \) is transcendental!)

Determine the integers \(n\) for which there exists a surjective homomorphism of symmetric groups \(S_{n+1}\longrightarrow S_n\).

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^2\longrightarrow R^3 \qquad \text {given by}\qquad \begin {pmatrix} x+1 & 0\\ 2x & x^2\\ 1 & 1 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Describe, up to similarity, all \(8\times 8\) matrices with minimal polynomial \(x^2(x-1)^3\).

Let \(a\) be an integer and \(p\) a prime number such that \(x^p-a\) is reducible in \(\mathbb {Q}[x]\). Prove that \(x^p-a\) has root in \(\mathbb {Q}\).

Determine, up to conjugacy, all real \(5\times 5\) matrices with minimal polynomial \((x-1)^2(x+1)\).

Let \(A\) be a square matrix with coefficients in \(\mathbb {C}\). Prove that there exist polynomials \(s(t)\) and \(n(t)\) with coefficients in \(\mathbb {C}\), such that \(s(A)\) is diagonalizable, \(n(A)\) is nilpotent, and \(A=s(A)+n(A)\).

Determine the minimal polynomial of \(\sin (2\pi /5)\) over \(\mathbb {Q}\).

Compute the characteristic polynomial, the minimal polynomial, and the Jordan form of the matrix \[ \begin {pmatrix} 3 & 1 & -1\\ 2 & 2 & -1\\ 2 & 2 & 0 \end {pmatrix}. \]

Let \(K[x]\) be a polynomial ring over a field \(K\), and let \(n\) be a positive integer. Classify, up to isomorphism, all finitely generated modules over the ring \(K[x]/(x^n)\).

Factor \(11x^5-11x^4+14x^2-21x+7\) into irreducible polynomials in \(\mathbb {Q}[x]\).

Prove that the polynomial \(x^5-x-1\) has no root in \(\mathbb {F}_9\), and that it is irreducible over \(\mathbb {F}_3\). Determine the integers \(n\) for which \(x^5-x-1\) is irreducible over \(\mathbb {F}_{3^n}\).

Prove that \(n\times n\) matrices \(A\) and \(B\) have the same characteristic polynomial if and only if the trace of \(A^k\) equals the trace of \(B^k\) for each \(k\ge 1\).

Let \(R\) be a Noetherian ring. Prove that any surjective ring homomorphism \(\varphi :R\longrightarrow R\) is an isomorphism.

Calculate the Galois group \(p(x):=x^3+3x+2\), viewed as a polynomial over \(\mathbb {Q}\).

Let \(K\) be a field and \(f(x)\in K[x]\) an irreducible polynomial. Let \(n\ge 2\) be an integer and set \(g(x):=f(x^n)\). Prove that if \(h(x)\) is an irreducible factor of \(g(x)\), then the degree of \(f\) divides the degree of \(h\).

Let \(R=\mathbb {Z}[x]\), set \(I=(x^3)\), and let \(U\) be the multiplicatively closed set \(\{n\in \mathbb {Z}\mid n\text { is odd}\}\). Find all the prime ideals in the ring \((U^{-1}R)/U^{-1}I\).

Let \(f(x)\) be a degree 5 polynomial in \(\mathbb {Q}[x]\) that is not solvable by radicals. Let \(L\) be its splitting field over \(\mathbb {Q}\).

1. Prove that there is at most one field \(K\) with \(\mathbb {Q}\subset K\subset L\) and \([K:\mathbb {Q}]=2\).
2. If \(\alpha ,\beta \) are irrational elements in \(L\) such that \(\alpha ^2\) and \(\beta ^2\) are rational, prove that \(\alpha \beta \) is rational.

Consider the polynomial ring \(\mathbb {Q}[x,y]\) where \(x,y\) are indeterminates over \(\mathbb {Q}\). Determine a finite generating set for the ideal of polynomials \(f(x,y)\) with \(f(i,i)=0\).

Suppose \(K\subset L\subset M\) are fields with \([M:L]=2=[L:K]\). Prove that \(M=K(\alpha )\), where \(\alpha \) is a root of an irreducible polynomial in \(K[x]\) of the form \(x^4+bx^2+c\).

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 \[ \operatorname {Tor}^{A}_{i}(\mathbb {F},\mathbb {F})\cong \operatorname {Ext}^{i}_{A}(\mathbb {F},\mathbb {F})\cong \bigwedge ^{i}(\mathbb {F}^{n}) \quad \text {for each } i\geq 0. \]

Compute the Galois group of the polynomial \((x^2-2)(x^3-2)(x^3-3)\) over the field \(\mathbb {Q}(\sqrt {-3})\).

Given an integer \(n\), prove that there exists a monic polynomial with integer coefficients whose Galois group over \(\mathbb {Q}\) is \(S_n\).

Let \(\alpha \) be a complex root of \(x^6+x^3+1\). Determine all field homomorphisms \(\mathbb {Q}(\alpha )\longrightarrow \mathbb {C}\).

Is the polynomial \(x^8+x+1\) irreducible in \(\mathbb {F}_2[x]\)?

Let \(f(x)\) be a monic polynomial with integer coefficients, such that \(f(\alpha )=0=f(2\alpha )\) for some complex number \(\alpha \). Prove that \(f(0)\ne 1\).

Show that if \(p\) is an odd prime, the polynomial \(x^{p^n}-x+1\) is irreducible over \(\mathbb {F}_p\) only when \(n=1\).

Let \(f(x)\in \mathbb {Q}[x]\) be a polynomial of degree \(n\ge 4\) and let \(K\) be a splitting field of \(f\) over \(\mathbb {Q}\). Suppose that \(\operatorname {Gal}(K/\mathbb {Q})\) is the symmetric group \(S_n\). If \(\alpha \in K\) is a root of \(f(x)\), show that \(\alpha ^n\notin \mathbb {Q}\).

Determine the splitting field and the Galois group for the polynomial \(x^3-2\) over \(\mathbb {Q}\).

Show that the polynomial \(x^4+1\) is reducible modulo every prime \(p\).

Let \(K\subseteq L\) be fields, and let \(f(x)\) be an irreducible polynomial in \(K[x]\). If there exists \(a\) in \(L\) with \(f(a)=0=f(a^2)\), prove that \(f(x)\) splits in \(L[x]\).

Show that the polynomial \(x^2+y^2-1\) is irreducible in \(\mathbb {Q}[x,y]\). Is it irreducible in \(\mathbb {C}[x,y]\)?

Set \(R=\mathbb {Q}[x]\), and consider the submodule \(M\) of \(R^2\) generated by the elements \((1-2x,-x^2)\) and \((1-x,x-x^2)\). Express \(R^2/M\) as a direct sum of cyclic modules.

Compute the number of monic irreducible polynomials of degree 3 over the field \(\mathbb {Z}_7\).

State and prove Hilbert’s basis theorem.

Determine for which integers \(q\) the polynomial \(X^3+1\) has three distinct roots in the field with \(q\) elements.

Let \(\mathfrak {a}\) be the ideal of the polynomial ring \(\mathbb {R}[x]\) that is generated by \(x^3-5x^2+6x-2\) and \(x^4-4x^3+3x^2-4x+2\). Is \(\mathfrak {a}\) a principal ideal? Is it prime? Is it maximal?

Suppose \(f(x)\in \mathbb {Q}[x]\) is an irreducible polynomial with complex roots \(\alpha _1,\ldots ,\alpha _n\). Prove that \(\alpha _i-\alpha _j\notin \mathbb {Q}\) for all \(i\ne j\).

Let \(K\) be a subfield of a field \(L\). If \(\sigma :L\longrightarrow L\) is a homomorphism that fixes all elements of \(K\), is \(\sigma \) necessarily an automorphism of \(L\)? Justify your answer.

Let \(K\) be a field and let \(f(x)\in K[x]\) be a monic polynomial of degree \(n\) with \(f(0)\ne 0\). Suppose \(f(x)\) has \(n\) distinct roots in its splitting field, and that the set of roots is closed under multiplication. Determine \(f(x)\).

Prove that \(\Phi _{2^n}(x)=x^{2^{n-1}}+1\) is irreducible in \(\mathbb {Q}[x]\).

Let \(I\subseteq \mathbb {Z}[x]\) be an ideal consisting of all polynomials \(a_0+a_1x+a_2x^2+\cdots \) such that 8 divides \(a_0\), 4 divides \(a_1\) and 2 divides all other coefficients. Find a finite set of generators of the ideal \(I\).

Let \(F\) be a field with \(\mathbb {Q}\subset F\subset \mathbb {C}\), where \(F/\mathbb {Q}\) is a finite Galois extension. Let \(\alpha \in F\) and let \(f(X)\in \mathbb {Q}[X]\) be its minimal monic polynomial. Assume that the absolute value \(|\alpha |\) is 1, and that \(\operatorname {Gal}(F/\mathbb {Q})\) is abelian.

1. Show that \(F\) is closed under complex conjugation.
2. Prove that \(|\beta |=1\) for every complex root \(\beta \) of \(f(X)\).
3. Writing \(f(X)=X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0\), show that \(|a_i|\le 2^n\) for all \(i\) with \(0\le i<n\).
4. Prove that \(F\) contains only finitely many algebraic integers having absolute value 1 and deduce that each of these is a root of unity.

Let \(\mathbb {Z}[x]\) be the ring of polynomials in \(x\) with integer coefficients. Let \(R\) denote the subring consisting of all polynomials having their coefficients of \(x\) and \(x^2\) equal to 0.

1. Show that \(\mathbb {Q}(x)\) is the field of fractions of \(R\).
2. Compute the integral closure of \(R\) in \(\mathbb {Q}(x)\).
3. Does there exist a polynomial \(g(x)\in R\) such that \(R\) is generated as a ring by 1 and \(g(x)\)?

Let \(R\) be a unique factorization domain with quotient field \(k\). Let \(f\) be an element of \(R[x]\). Show that if \(f\) can be factored as a product of lower-degree polynomials in \(k[x]\), then \(f\) can be factored as a product of lower-degree polynomials in \(R[x]\).

Determine the group of units in the ring \(\mathbb {Q}[x]/((x^2+1)^2)\).

Prove Gauss’ Lemma over the integers: if \(h(x)\in \mathbb {Z}[x]\) such that \(h(x)=f(x)g(x)\) for nonconstant polynomials \(f(x),g(x)\in \mathbb {Q}[x]\), prove that there exist rational numbers \(a\) and \(b\) so that \(F(x)=af(x)\) and \(G(x)=bg(x)\) are in \(\mathbb {Z}[x]\) and so that \(h(x)=F(x)G(x)\).

Let \(G\) be the Galois group of the polynomial \(x^4-5\) over the field \(F\). Determine \(G\) in each of the following cases:

1. \(F\) is the real numbers;
2. \(F\) is the rational numbers;
3. \(F\) is the integers modulo 3.

Let \(A\) be an \(n\)-by-\(n\) diagonal matrix with coefficients in a field \(F\). Suppose the characteristic polynomial of \(A\) has the form \[ (x-c_1)^{d_1}(x-c_2)^{d_2}\cdots (x-c_k)^{d_k} \] with \(c_1,\ldots ,c_k\) distinct. Let \(V\) denote the space of all \(n\)-by-\(n\) matrices over \(F\) which commute with \(A\). Compute the dimension of \(V\) as a vector space over \(F\).

Suppose \(R\) is a commutative Noetherian ring. Let \(R[x]\) denote the ring of polynomial with coefficients in \(R\). Prove that \(R[x]\) is Noetherian.

Suppose \(R\) is a commutative Noetherian ring. Let \(R[[x]]\) denote the ring of formal power series with coefficients in \(R\). Prove that \(R[[x]]\) is Noetherian.

Compute the Galois group of the polynomial \(x^4-2\) over each of these fields: (a) the complex numbers (b) the reals, (c) the rationals, and (d) the integers modulo 5.

Let \(R\) be the ring of polynomials in one variable over a field \(F\). Prove that \(R\) has infinitely many maximal ideals.

Let \(G\) be the Galois group of the polynomial \(x^3-3x^2-3\) over the field \(F\). Determine \(G\) in each of the following cases:

1. \(F\) is the complex numbers.
2. \(F\) is the real numbers.
3. \(F\) is the rational numbers.
4. \(F\) is the integers modulo 2.

Prove that there is a rational polynomial whose Galois group over the rationals is the direct product of two groups of order 3.

Determine for which fields the polynomial \(X^3-X+1\) has a multiple root.

Show that a polynomial \(f(X)\) of degree \(n\) over the field \(\mathbb {Z}/p\mathbb {Z}\) is irreducible if and only if \(f(X)\) is relatively prime to the polynomial \(X^{p^m}-X\) for all \(m\) less than or equal to \(n/2\).

Let \(R\) be a unique factorization domain with quotient field \(K\). Let \(f(X)\) be a polynomial in \(R[X]\). Show that if \(f(X)\) can be factored into polynomials of lower degree in \(K[X]\), then \(f(X)\) can be factored into polynomials of lower degree in \(R[X]\).

Let \(G\) be the Galois group of the polynomial \(x^3+2x-2\) over the field \(F\). Determine \(G\) in each of the following cases:

1. \(F\) is the complex numbers.
2. \(F\) is the real numbers.
3. \(F\) is the rational numbers.
4. \(F\) is the integers modulo 3.

Prove that if \(A\) is a commutative Noetherian ring, then \(A[X_1,\ldots ,X_n]\) (the polynomial ring in \(n\) variables over \(A\)) is Noetherian.

Let \(M\) be an \(n\) by \(n\) matrix with entries in \(\mathbb {C}\). Assume that there is a polynomial \(f(X)\) in \(\mathbb {Q}[X]\) which is irreducible over \(\mathbb {Q}\) such that \(f(M)=0\) (i.e. it is the zero matrix). Show that \(M\) is diagonalizable over \(\mathbb {C}\).

State and prove the Hilbert Basis Theorem.

Let \(V\) be a finite dimensional vector space over an algebraically closed field \(F\). Let \(T\) be a linear transformation of \(V\) into itself, let \(\lambda \) be an eigenvalue of multiplicity \(m\) of \(T\) (i.e., \(m\) is the multiplicity of \(\lambda \) as a root of the characteristic polynomial of \(T\)), and let \(U\) be the subspace \(U=\{v\in V\mid T(v)=\lambda v\}\). Prove that the dimension of \(U\) is \(\le m\).

Prove that there is a polynomial \(p(x)\) with rational coefficients such that the Galois group of \(p(x)\) over the field of rational numbers is cyclic of order 5.

Prove the Cayley-Hamilton Theorem (the theorem that states that for every matrix \(A\) we have \(f(A)=0\), where \(f(X)\) is the characteristic polynomial of \(A\)).

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.

If \(R\) is a left noetherian ring, then \(R[x]\) is left noetherian.