Let \(K\) be the field \(\mathbb Q(z)\) of rational functions in a variable \(z\) with coeffiecients in the rational field \(\mathbb Q\). Let \(n\) be a positive integer. Consider the polynomial \(x^n-z\in K[x]\).
Show that the polynomial \(x^n-z\) is irreducible over \(K\).
Describe the splitting field of \(x^n-z\) over \(K\).
Determine the Galois group of the splitting field of \(x^5-z\) over the field \(K\).
Solution
Goal. For \(K = \QQ(z)\), show \(x^n - z\) is irreducible over \(K\), describe its splitting field, and compute the Galois group for \(n = 5\).
1. \(x^n - z\) is irreducible over \(K = \QQ(z)\).
1.1. \(z\) is prime in \(\QQ[z]\), and \(x^n - z\) is Eisenstein at the prime \(z\).
\(z\) is irreducible in the UFD \(\QQ[z]\); the polynomial \(x^n - z\) has leading coefficient \(1\), all non-leading coefficients \(0\) (divisible by \(z\)), and constant term \(-z\) not divisible by \(z^2\).
1.2. Hence \(x^n - z\) is irreducible in \(\QQ[z][x] = \QQ[z, x]\).
Eisenstein’s criterion.
1.3. Hence \(x^n - z\) is irreducible in \(\QQ(z)[x] = K[x]\).
Gauss’s lemma — irreducibility in \(\QQ[z][x]\) passes to the fraction field \(\QQ(z)\).
2. The splitting field of \(x^n - z\) over \(K\).
2.1. Adjoin a root \(\alpha\) with \(\alpha^n = z\); then \(x^n - z = \prod_{j=0}^{n-1} (x - \zeta_n^j \alpha)\).
the roots are \(\zeta_n^j \alpha\) for \(j = 0, \dots, n-1\), where \(\zeta_n\) is a primitive \(n\)-th root of unity.
2.2. The splitting field is \(K(\alpha, \zeta_n) = \QQ(z)(\alpha, \zeta_n)\).
it is generated by one root \(\alpha\) and the \(n\)-th roots of unity \(\zeta_n\).
3. The Galois group of the splitting field of \(x^5 - z\) over \(K\).
3.1. The splitting field is \(L = K(\alpha, \zeta_5)\) with \([K(\alpha) : K] = 5\) and \([K(\zeta_5) : K] = 4\).
\(x^5 - z\) is irreducible of degree \(5\) by 1, so \([K(\alpha):K] = 5\); the cyclotomic polynomial \(\Phi_5\) has degree \(4\) and is irreducible over \(\QQ\), hence over \(K\) (since \(K \cap \QQ(\zeta_5) = \QQ\)).
3.2. \([L : K] = 20\).
\(K(\alpha)\) and \(K(\zeta_5)\) are linearly disjoint over \(K\) (their degrees \(5\) and \(4\) are coprime), so \([L:K] = 5 \cdot 4 = 20\).
3.3. \(\Gal(L/K) \cong \ZZ/5 \rtimes (\ZZ/5)^\times \cong \ZZ/5 \rtimes \ZZ/4\).
an automorphism sends \(\alpha \mapsto \zeta_5^a \alpha\) (\(a \in \ZZ/5\)) and \(\zeta_5 \mapsto \zeta_5^b\) (\(b \in (\ZZ/5)^\times\)), giving the semidirect product \(\ZZ/5 \rtimes (\ZZ/5)^\times\) of order \(20\).
4. Q.E.D.
1, 2, and 3 answer the three parts.