Galois Group - Examples

Examples

In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F.

• Gal(F/F) is the trivial group that has a single element, namely the identity automorphism.
• Gal(C/R) has two elements, the identity automorphism and the complex conjugation automorphism.
• Aut(R/Q) is trivial. Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity.
• Aut(C/Q) is an infinite group.
• Gal(Q(√2)/Q) has two elements, the identity automorphism and the automorphism which exchanges √2 and −√2.
• Consider the field K = Q(³√2). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension, since the other two cube roots of 2 (both complex) are missing from the extension — in other words K is not a splitting field.
• Consider now L = Q(³√2, ω), where ω is a primitive third root of unity. The group Gal(L/Q) is isomorphic to S₃, the dihedral group of order 6, and L is in fact the splitting field of x3 − 2 over Q.
• If q is a prime power, and if F = GF(q) and E = GF(qn) denote the Galois fields of order q and qn respectively, then Gal(E/F) is cyclic of order n.
• If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two non-real roots, then the Galois group of f is the full symmetric group S_p.