Galois Theory

Let \(\zeta _8 = \exp (2\pi i/8)\), complex eight root of one. Use Galois theory to determine all subfields of \(\mathbb {Q}(\zeta _8)\).

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.

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 \(S_p\).

Find the Galois group of \(x^8-2 \in \mathbb {Q}[x]\). In particular, prove that it has order 16 and is not the dihedral group.

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

Let \(K = \mathbb {Q}[\sqrt {2},\sqrt {3},\sqrt {5}]\). Show that \(K\) is Galois over \(\mathbb {Q}\) and identify \(\operatorname {Gal}(K/\mathbb {Q})\) up to isomorphism.

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 \(E \ne F[\alpha ^2]\).

Find a complex number \(\eta \) such that \(\mathbb {Q}(\eta )/\mathbb {Q}\) is Galois with Galois group isomorphic to the dihedral group of order 8.

Suppose \(f(x) \in \mathbb {Q}[x]\) is a polynomial with Galois group of order \(594 = 2 \cdot 3^3 \cdot 11\). Show that \(f\) is solvable in radicals.

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 \(2^{1/4}\) 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.

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

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.

What is the Galois closure of \(\mathbb {Q}(\sqrt {1+\sqrt {2}})/\mathbb {Q}\)?

Find the Galois group of \(f(x)=x^4+2x^2+x+3\).

Determine the Galois group of \(x^6+3\) over \(\mathbb {Q}\).

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?

Determine the Galois group of \(x^8-2\) over \(\mathbb {Q}\).

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

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

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

Let \(E:=\mathbb {Q}(\zeta +\zeta ^{-1})\) where \(\zeta \in \mathbb {C}\) is a primitive \(n\)th root of 1, for some integer \(n\ge 3\). Determine the Galois group of \(E\) over \(\mathbb {Q}\).

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.

Determine the Galois group of \(x^4-16x^2+4\) 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 \(L[x]\).

Show that \(K=\mathbb {Q}(\sqrt {1+\sqrt {3}})\) is not Galois over \(\mathbb {Q}\), and compute \([K:\mathbb {Q}]\).

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

Compute the Galois group of \(x^4-2\) over \(\mathbb {Q}\).

Prove that \(\mathbb {Q}(\sqrt {5+\sqrt {5}})\) is Galois over \(\mathbb {Q}\), and compute the Galois group.

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 \(\mathbb {F}\) 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 \(\alpha \) be a complex root of \(x^6+3\). Set \(K=\mathbb {Q}(\alpha )\).

1. Prove that \(K\) contains a primitive 6-th root of unity.
2. Compute the Galois group of \(K\) over \(\mathbb {Q}\).

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

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

What is the Galois group of \(x^4-5\) over:

1. \(\mathbb {Q}\);
2. \(\mathbb {Q}(\sqrt {5})\);
3. \(\mathbb {Q}(i)\).

Show that the extension \(\mathbb {Q}(\sqrt {2},\sqrt {3})\) is Galois over \(\mathbb {Q}\), and determine the Galois group.

Consider the automorphisms \(\sigma ,\tau \) of \(\mathbb {Q}(x)\) with \(\sigma :x\mapsto 1/x\) and \(\tau :x\mapsto 1-x\). What is the order of the group generated by these two elements? Determine the group.

For \(p\) a prime, determine the Galois group of \(x^p-2\) over \(\mathbb {Q}\). What is its order? Is it abelian?

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.

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.

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

Let \(L=\mathbb {Q}(e^{2\pi i/8})\). Determine all subgroups of the Galois group \(\operatorname {Gal}(L/\mathbb {Q})\), and the corresponding intermediate fields.

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 \(K\).

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. \(\mathbb {F}_7\)

Determine the Galois group of \(x^4-2\) over each of the following fields: the complex numbers; the real numbers; the rational numbers; the integers modulo 5.

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.

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

Let \(G_F\) denote the Galois group of \(x^3-3x^2-3\) over the field \(F\). Determine \(G_F\) when:

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

Prove that the Galois group of \(x^3+x-1\) over \(\mathbb {Q}\) is the symmetric group on three letters.

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 \(G\) 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.

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.

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

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.

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 \(\alpha ^p-\alpha \in F\).

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 \(K\) be an arbitrary field, and let \(L\) be an extension of \(K\) obtained by adjoining a root of the polynomial \(X^n-1\) for some integer \(n\ge 1\). Show that \(L\) is Galois over \(K\) with abelian Galois group.

Let \(k\) be a field containing a primitive \(n^{\mathrm {th}}\) root of unity. Show that for any \(a\in k\) the Galois group of \(X^n-a\) over \(k\) is abelian.

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.

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

Prove that the Galois group of \(X^3-X-1\) over \(\mathbb {Q}\) is \(S_3\), the symmetric group on three elements.

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.

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

Let \(K\) be a field of characteristic zero. Let \(L\) be the splitting field of the polynomial \(X^n-a\) for some \(a\in K\) and some integer \(n\ge 1\). Show that the Galois group of \(L\) over \(K\) is solvable.

Show that if the field \(F\) contains a primitive \(n^{\mathrm {th}}\)-root of unity, then the Galois group of the polynomial \(x^n-a\in F[x]\), \(a\in F\), is abelian.