Irreducible Polynomials

Let \(G\) be a finite group. Let \(V_1\) and \(V_2\) be two non-isomorphic irreducible complex representations. Let \(V = V_1 \oplus V_2\). Compute the algebra of linear transformations of \(V\) intertwining the action of \(G\): \[ \operatorname {Hom}_G(V,V). \]

Let \(K\) be an arbitrary field. Prove that the polynomial ring \(K[x]\) has infinitely many prime ideals.

Show that if \(L/K\) is Galois and \(f \in K[x]\) is monic irreducible, then every irreducible factor of \(f\) in \(L[x]\) has the same degree.

Find a complete list of irreducible representations of the alternating group \(A_4\).

How many irreducible polynomials of degree 12 in \(\mathbb {F}_p[x]\) are there?

Prove that the Galois group of an irreducible polynomial \(f(x) \in \mathbb {Q}[x]\) of degree \(p\) (a prime) with one pair of complex conjugate roots and \(p-2\) real roots is the full symmetric group \(S_p\).

Let \(f(x)\) be an irreducible polynomial in \(\mathbb {Q}[x]\), with splitting field \(K\). Suppose \(\operatorname {Gal}(K/\mathbb {Q})\) is abelian. Prove or disprove: \(K = \mathbb {Q}(\alpha )\) for each root \(\alpha \) of \(f(x)\) in \(K\).

Find a degree 5 monic polynomial with integer coefficients that is irreducible in \(\mathbb {Q}[x]\) but does not satisfy the Eisenstein criterion for any prime \(p\).

For \(p\) a prime number, show that \(x^{p^n}-x \in \mathbb {F}_p[x]\) is the product of all irreducible polynomials of degree dividing \(n\) in \(\mathbb {F}_p[x]\).

Recall that the standard representation of \(S_n\) is the subspace

with \(S_n\) acting by permutation of the coordinates. Show that \(\wedge ^2 V_5\) is an irreducible representation of \(S_5\).

Show that if \(L/K\) is Galois and \(f \in K[x]\) is monic irreducible, then every irreducible factor of \(f\) in \(L[x]\) has the same degree.

Let \(K\) be an algebraically closed field. Let \(f\in K[x,y]\) be irreducible and let \(g\in K[x,y]\) be such that \(f\) does not divide \(g\). Prove that there is a point \((k_1,k_2)\in K^2\) such that \[ f(k_1,k_2)=0\text { and }g(k_1,k_2)\ne 0. \]

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

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 \(M\).

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 \(K\) 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 \(L=K(\alpha )\).

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

Determine the irreducible factorization of \(x^5-1\) over the field \(\mathbb {F}_{19}\).

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 \(\mathbb {Q}\).

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 minimal polynomial of \(\sqrt {4+\sqrt {7}}\) over \(\mathbb {Q}\). Determine the Galois group of this polynomial.

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

Let \(K=\mathbb {Q}(\pi )\), where \(\pi \) is the ratio of the circumference of a circle to its diameter. You may assume without proof that \(\pi \) is transcendental over \(\mathbb {Q}\).

1. Show that \(f(x)=x^3-\pi \) is irreducible over \(K\).
2. Let \(L\) denote the splitting field of \(f(x)\) over \(K\). Compute \([L:K]\) and \(\operatorname {Gal}(L/K)\).

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

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

Let \(L/K\) be a Galois extension of fields of characteristic zero. Suppose \(f(x)\) is an irreducible polynomial of degree 5 in \(K[x]\). If \(f(x)\) has no roots in \(L\), prove that it is irreducible in \(L[x]\).

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

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 \(k\ge 1\).

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 \(p\) be a prime number. Prove that if \(x^{p^n}-x+1\) is irreducible over \(\mathbb {F}_p\), then \(n=1\) or \(n=2=p\).

(Hint: Note that if \(\alpha \) is a root of the equation, then \(\alpha ^p-\alpha \) is in \(\mathbb {F}_{p^n}\).)

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 \(x^4+bx^2+c\).

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 \(n=1\).

Show that \(f(x)=x^4+4x^2+2\) is irreducible over \(\mathbb {Q}\), and find its Galois group 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 \(\mathbb {C}[x,y]\)?

Prove that \(x^4+1\) is reducible over any field of positive characteristic.

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

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

State and prove the Eisenstein Irreducibility Criterion.

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 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 \(G\) be the group of symmetries of an equilateral triangle. Show that the 2 dimensional representation (as symmetries of the triangle) is irreducible by calculating the character table.

Let \(D_8\) be the group of symmetries of a square. Show that the 2 dimensional representation (as symmetries of the square) is irreducible by calculating the character table.

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 \(p\) be a prime number and \(\mathbb {F}_p\) be the field with \(p\) elements. Let \(f(x)=x^p-x-1\in \mathbb {F}_p[x]\) and let \(K\) be the splitting field of \(f(x)\). Show that:

1. if \(\alpha \) is a root of \(f(x)\), then \(\alpha +1\) is a root of \(f(x)\),
2. \(K=\mathbb {F}_p(\alpha )\),
3. the Frobenius automorphism \(\sigma (x)=x^p\) is a non-trivial element of \(\operatorname {Gal}(K/\mathbb {F}_p)\),
4. \(f(x)\in \mathbb {F}_p[x]\) is irreducible.

Let \(\alpha \) be a primitive fifth root of 1 and \(f(x)=x^5+5x^2+4\). Show that \(f(x)\) is irreducible in \(\mathbb {Q}[\alpha ]\).

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.

Prove or disprove that \(f(x)=x^5-x+1\) is irreducible over the rationals.

Prove or disprove that \(f(x)=x^4+x^3+x^2+6x+1\) is irreducible over \(\mathbb {Q}\).

Prove or disprove that \(x^5-x+1\) is irreducible over the rationals.

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 a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following statements is true about \(M\) considered as an abelian group:

1. \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all rational numbers.
2. \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order \(p\) for some prime \(p\).

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.

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

Let \(E\) be a finite separable extension of a field \(K\) and let \(F\) be any finite field extension of \(K\). Show that \(E\otimes _K F\) is a finite product of fields.

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 a finite group. Prove the following version of Schur’s Lemma: if \(\rho \colon G\to M_n(\mathbb {C})\) is a finite dimensional irreducible representation of \(G\) over the complex numbers, then every element of \(M_n(\mathbb {C})\) that commutes with \(\rho (g)\) for all \(g\in G\) is a scalar matrix.

Let \(f(x)\) be a polynomial of degree \(n>2\) over \(\mathbb {Q}\), the field of rational numbers. Let \(K\) be the splitting field of \(f(x)\) over \(\mathbb {Q}\) and suppose that the Galois group of \(f(x)\) over \(\mathbb {Q}\) is \(S_n\), the group of all permutations of \(n\) objects. Prove the following:

1. \(f(x)\) is irreducible over \(\mathbb {Q}\).
2. Let \(\alpha \) be a root of \(f(x)\) in \(K\) and let \(\sigma \) be an automorphism of the field \(\mathbb {Q}(\alpha )\). Prove that \(\sigma =1\).

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.

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.

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 \(\mathbb {C}\).

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 \(A\)).

Let \(R\) be a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following statements is true about \(M\) considered as an abelian group:

1. \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all rational numbers.
2. \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order \(p\) for some prime \(p\).

If \(p(x)\in F[x]\) where \(F\) is a finite field, then \(p(x)\) is separable over \(F[x]\).