Splitting Fields

Let \(K\) be the splitting field of \(x^4-2x^2+4\) over \(\mathbb {Q}\). Determine \(\operatorname {Gal}(K/\mathbb {Q})\).

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

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 \(K\).

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?

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.

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

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 \(\operatorname {Gal}(L/K)\).

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 \(f(x)\) 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.

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 \(\mathbb {Q}\).

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

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 \(f(x)\).

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 \(p\) 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 \(f(x)\) 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 \(K\) 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.