Let \(K\) be the splitting field of \(X^4 - 12\) over \(\mathbb{Q}\).
Give an explicit description of \(K\) and compute its degree over \(\mathbb{Q}\).
What is the Galois group of \(K\) over \(\mathbb{Q}\)? You should identify the group as well as its action on the roots of \(X^4 - 12\).
Solution
Let \(\alpha=12^{1/4}>0\) be the real positive fourth root of \(12\).
1. The roots of \(X^4-12\) are \(r_j=i^j\alpha\) for \(j\in\ZZ/4\ZZ\), that is, \(\alpha, i\alpha, -\alpha, -i\alpha\).
Each \(r_j\) satisfies \(r_j^4=i^{4j}\alpha^4=12\), and the four numbers are distinct, so they are all the roots of the quartic.
2. \(K=\QQ(\alpha,i)\).
The field \(\QQ(\alpha,i)\) contains every \(r_j\). Conversely, \(K\) contains \(\alpha=r_0\) and \(i=r_1/r_0\).
3. \([\QQ(\alpha):\QQ]=4\).
Since \(3\mid12\) and \(9\nmid12\), the polynomial \(X^4-12\) is Eisenstein at \(3\), hence irreducible over \(\QQ\), and it is the minimal polynomial of \(\alpha\).
4. \([K:\QQ]=8\).
The field \(\QQ(\alpha)\) lies in \(\RR\), so \(i\notin\QQ(\alpha)\) and \([K:\QQ(\alpha)]=2\), with minimal polynomial \(X^2+1\). The tower law and step 3 give \([K:\QQ]=4\cdot2=8\).
5. \(\Gal(K/\QQ)\cong D_8\), the dihedral group of order \(8\). It is generated by \(\sigma\) with \(\sigma(\alpha)=i\alpha\), \(\sigma(i)=i\), and \(\tau\) with \(\tau(\alpha)=\alpha\), \(\tau(i)=-i\). On the roots, \(\sigma(r_j)=r_{j+1}\) and \(\tau(r_j)=r_{-j}\): \(\sigma\) is a rotation and \(\tau\) a reflection of the square with vertices \(r_0,r_1,r_2,r_3\).
By step 4, \([K:\QQ(i)]=4\), so \(X^4-12\) is irreducible over \(\QQ(i)\) and there is a \(\QQ(i)\)-automorphism \(\sigma\) of \(K\) with \(\sigma(\alpha)=i\alpha\). Complex conjugation restricts to \(\tau\). Then \(\sigma(r_j)=\sigma(i)^j\sigma(\alpha)=i^{j+1}\alpha=r_{j+1}\) and \(\tau(r_j)=(-i)^j\alpha=r_{-j}\). Hence \(\sigma\) has order \(4\), \(\tau\) has order \(2\), and \(\tau\sigma\tau=\sigma^{-1}\), since both sides send \(r_j\) to \(r_{j-1}\). Every power of \(\sigma\) fixes \(i\) and \(\tau\) does not, so \(\tau\notin\langle\sigma\rangle\) and \(\langle\sigma,\tau\rangle\) has order \(8=\abs{\Gal(K/\QQ)}\). These are the defining relations of \(D_8\).
6. Q.E.D.
Steps 2 and 4 answer part (a), and step 5 answers part (b).