Finite Fields

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

Determine the splitting field and the Galois group of \(x^9+1\) over \(\mathbb {F}_2\).

Let \(F\) be a field of characteristic zero and let \(E/F\) be a finite Galois extension. Suppose that \(E = F[\alpha ]\). Suppose that there exists \(\sigma \in G = \operatorname {Gal}(E/F)\) with \(\sigma (\alpha )=-\alpha \). Prove that \(E \ne F[\alpha ^2]\).

Let \(L\) be the splitting field of \((x^3+2x+1)(x^3+x^2+2)(x^2+1)\) over \(\mathbb {F}_3\). How many proper subfields does \(L\) have?

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 \(\mathbb {F}_p[x]\).

Suppose \(R=\mathbb {F}_3[x]\). Let \(M\) be the module generated by three elements \(a,b,c\) subject to the three relations \(-xa+x^2b+(x^2-1)c=0\), \(xb+xc=0\) and \(xb+x^2c=0\). Find a direct sum of cyclic \(R\)-modules isomorphic to \(M\).

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 \(\sigma \) be the automorphism of \(\mathbb {F}_5(t)\) sending \(t\) to \(t+1\). Let \(G\) be the subgroup of \(\operatorname {Aut}(\mathbb {F}_5(t))\) generated by \(\sigma \). Find an element \(s\in \mathbb {F}_5(t)\) such that \[ \mathbb {F}_5(t)^G=\mathbb {F}_5(s). \]

Determine the splitting field of the polynomial \(x^p-x-a\) over \(\mathbb {F}_p\) where \(0\ne a\in \mathbb {F}_p\). Show that the Galois group is cyclic.

Suppose that \(k\) is the field with 3 elements and let \(R=k[x]\). Identify nine different (up to isomorphism) \(R\)-modules \(M\) such that \(|M|=9\).

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

Suppose \(R=\mathbb {F}_2[x,y,z]\). Compute \(\operatorname {Tor}^R_1(R/(x,y),R/(y,z))\).

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

Let \(p\) be a prime number and \(n\) a positive integer. How many elements \(\alpha \) are in \(\mathbb {F}_{p^n}\) such that \(\mathbb {F}_p(\alpha )=\mathbb {F}_{p^6}\)?

Suppose \(A\) is a finitely generated abelian group such that \(A\otimes _{\mathbb {Z}}\mathbb {F}=0\) for all finite fields \(\mathbb {F}\). Show that \(A=0\).

Let \(\mathbb {F}_5\) be the field with 5 elements. Construct an explicit isomorphism between the fields \(\mathbb {F}_5[x]/(x^2-2)\) and \(\mathbb {F}_5[x]/(x^2-3)\).

Let \(p\) be a prime number, \(n\) a positive integer, and \(V\) the field \(\mathbb {F}_{p^n}\) viewed as a vector space over \(\mathbb {F}_p\). Prove that the Frobenius map \(\varphi :V\longrightarrow V\), assigning \(v\) to \(v^p\), is diagonalizable over \(\mathbb {F}_p\) if and only if \(n\) divides \(p-1\).

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 \(\mathbb {F}_{3^n}\).

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 \(\mathbb {F}_{p^n}\).)

Prove that \(\mathbb {F}=\mathbb {Z}[t]/\langle 3, t^3-t^2+1\rangle \) is a field. Find the number of solutions of \(x^{13}+1=0\) in \(\mathbb {F}\), and also the number of solutions of \(x^{13}-1=0\) in \(\mathbb {F}\).

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 finite field with \(q\) elements. Determine the number of \(2\times 2\) nilpotent matrices with entries from \(\mathbb {F}\).

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 \(M\) be an \(n\times n\) matrix with entries from a field \(\mathbb {F}\) such that \(M^3=I\). Is \(M\) necessarily diagonalizable if (a) \(\mathbb {F}=\mathbb {Q}(\sqrt {3})\), (b) \(\mathbb {F}=\mathbb {Q}(i)\), (c) \(\mathbb {F}=\mathbb {F}_3\)?

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

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 \(\mathbb {R}\).

Prove that each element of a finite field can be written as a sum of two squares.

Determine the number of \(2\times 2\) nilpotent matrices over a finite field \(\mathbb {F}_q\).

Let \(\zeta \) be a primitive 16-th root of unity over a field \(K\). Determine \([K(\zeta ):K]\) when \(K\) is:

1. \(\mathbb {F}_7\)
2. \(\mathbb {F}_9\)
3. \(\mathbb {F}_{17}\)

Explain (preferably in a sentence!) why \(\mathbb {F}_3[x]/(x^2-2)\) and \(\mathbb {F}_3[x]/(x^2-2x-1)\) are isomorphic. Then construct an explicit isomorphism.

Determine the number of Sylow \(p\)-subgroups of \(GL_2(\mathbb {F}_p)\).

Determine representatives for the conjugacy classes in \(GL_3(\mathbb {F}_2)\).

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 \(\phi \) denote the Frobenius map \(x\mapsto x^p\) on the finite field \(\mathbb {F}_{p^n}\). Determine the Jordan canonical form (over \(\overline {\mathbb {F}}_p\)) for \(\phi \) regarded as an \(\mathbb {F}_p\)-linear transformation of \(\mathbb {F}_{p^n}\).

Prove that the number of Sylow \(p\)-subgroups of \(GL_2(\mathbb {F}_p)\) is \(p+1\).

Determine the number of 5-Sylow subgroups of \(\operatorname {SL}_2(\mathbb {F}_5)\).

Find the degree of the splitting field of \(x^6-3\) over (i) \(\mathbb {Q}(\sqrt {-3})\) and (ii) \(\mathbb {F}_5\).

Let \(E\subset F\) be a finite Galois extension of fields. Suppose that there is an element \(\alpha \) in \(F\) such that \(\alpha \notin E\) and such that \(\alpha \) is in every proper extension of \(E\) contained in \(F\). Show that the Galois group of \(F\) over \(E\) is cyclic of prime power order.

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

Let \(\mathbb {F}\) be a field with 81 elements. What is the number of roots of \(x^{25}-1=0\) in \(\mathbb {F}\)?

Let \(\mathbb {F}_q\) be a finite field of order \(q\). Compute the order of \(GL_n(\mathbb {F}_q)\), the group of invertible \(n\times n\) matrices with coefficients in \(\mathbb {F}_q\).

Compute the Galois group \(G(E/F)\) where \(E\) is the splitting field of the polynomial \(x^3-2\) and \(F\) is

1. \(\mathbb {Q}\)
2. \(\mathbb {R}\)
3. \(\mathbb {F}_5\)
4. \(\mathbb {F}_7\)

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 \(V\) be a finite-dimensional vector space over a field of characteristic \(p>0\). Let \(T:V\longrightarrow V\) be a linear transformation with \(T^{p^n}=\operatorname {Id}\) for some \(n\). Prove that there is a nonzero vector \(v\in V\) with \(Tv=v\).

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 \(K\) 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 \(\theta \in K\)).

Let \(V\) be a finite dimensional vector space over a field \(F\) and \(T:V\longrightarrow V\) be a linear transformation. Show that there is an integer \(n>0\) such that \[ V\cong \operatorname {Ker}(T^n)\oplus \operatorname {Im}(T^n). \]

Let \(K\) be a finite field.

1. Show that \(|K|=p^n\) for some prime \(p\).
2. Show that \(K^*=K-\{0\}\) is a cyclic group with respect to multiplication.

Let \(F\) be a field such that the multiplicative group of \(F\) is cyclic. Prove that \(F\) is a finite field.

Let \(V\) be a finite-dimensional vector space over a field \(k\), and let \(A\) and \(B\) be diagonalizable linear operators on \(V\) such that \(AB=BA\). Prove that \(A\) and \(B\) are simultaneously diagonalizable; i.e. that there is a basis for \(V\) consisting of eigenvectors of both \(A\) and \(B\).

Let \(F\) be a finite field, and suppose that the subfield of \(F\) generated by \(\{x^3\mid x\in F\}\) is different from \(F\). Show that the cardinality of \(F\) is 4.

Let \(V\) be a finite dimensional vector space over the field of rational numbers and let \(T\) be a linear transformation of \(V\) into itself such that \(T^5\) is the identity mapping. If there is no nonzero vector in \(V\) which \(T\) maps to itself, prove that the dimension of \(V\) is a multiple of 4.

1. Suppose \(K\) is a field of prime order and \(E\) is the algebraic closure of \(K\). If \(F\) is any subfield of \(E\), prove that the multiplicative group of \(F\) is periodic, i.e., every element of the multiplicative group has finite order.
2. Suppose \(F\) is a field whose multiplicative group is periodic. Prove that for some prime \(p\), \(F\) is isomorphic to a subfield of the algebraic closure of the field of order \(p\).

Let \(E\subset F\) be an extension of finite fields. Let \(q\) be the number of elements of \(E\), and let \(\phi \colon F\to F\) be defined by \(\phi (a)=a^q\). Prove that \(F\) is a Galois extension of \(E\) and that \(\phi \) generates its Galois group.

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

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.

Suppose \(F\) is a field and \(K=F(\alpha )\) is a finite Galois extension of \(F\). Assume that the Galois group of \(K\) over \(F\) is cyclic generated by an automorphism \(\sigma \) where \(\alpha ^\sigma =\alpha +1\). Prove that \(F\) has characteristic \(p\ne 0\) and that \(\alpha ^p-\alpha \in F\).

If \(H\) is a finite group, prove that there exists a field \(k\) and a Galois extension of \(k\) with Galois group \(H\).

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 \(\le m\).

Let \(F\) be a field such that the multiplicative group of \(F\) is cyclic. Prove that \(F\) is a finite field.

Let \(K\subseteq L\) be finite fields. Prove that the extension \(L\) of \(K\) is Galois with cyclic Galois group.

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

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

A finite subgroup of the multiplicative group of nonzero elements of a field is cyclic.