In mathematics, more specifically in the area of abstract algebra
known as Galois theory, the
Contents 1 Definition 2 Examples 3 Properties 4 See also 5 Notes 6 References 7 External links Definition[edit]
Suppose that E is an extension of the field F (written as E/F and read
"E over F"). An automorphism of E/F is defined to be an automorphism
of E that fixes F pointwise. In other words, an automorphism of E/F is
an isomorphism α from E to E such that α(x) = x for each x ∈ F.
The set of all automorphisms of E/F forms a group with the operation
of function composition. This group is sometimes denoted by Aut(E/F).
If E/F is a Galois extension, then Aut(E/F) is called the "Galois
group of (the extension) E over F, and is usually denoted by
Gal(E/F).[1]
If E/F is not a Galois extension, then the
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.[2]
Aut(R/Q) is trivial. Indeed, it can be shown that any automorphism of
R 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 complex cube roots of 2 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 cube 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 and generated by the Frobenius homomorphism.
If f is an irreducible polynomial of prime degree p with rational
coefficients and exactly two nonreal roots, then the
Properties[edit]
The significance of an extension being Galois is that it obeys the
fundamental theorem of Galois theory: the closed (with respect to the
Krull topology) subgroups of the
Absolute Galois group Notes[edit] ^ Some authors refer to Aut(E/F) as the
References[edit] Jacobson, Nathan (2009) [1985]. Basic Algebra I (2nd ed.). Dover Publications. ISBN 978-0-486-47189-1. Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 External links[edit] Hazewinkel, Michiel, ed. (2001) [1994], "Galois group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 "Galois Groups". Ma |