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§1. Galois Theory: Theorems of Dedekind and Artin 145

(1.3) Remark. Let K be a finite Galois extension of F with Galois group Gal(K/F).
The function which assigns to a subgroup H of Gal(K/F) the subfield Fix(H) of K
containing F is a bijection from the set of subgroups of Gal(K/F) onto the set of
subfields L of K which contain F. The inverse function is the function which assigns
to such a subfield L of K the subgroup G L of all s in Gal(K/F) which restrict to
the identity on L. These functions are inclusion reversing, and K over L is a Galois
extension with G L = Gal(K/L). Further, the extension L over F is a Galois exten-
sion if and only if Gal(K/L) is a normal subgroup of Gal(K/F), and in this case the
quotient group Gal(K/F)/Gal(K/L) is isomorphic to the Galois group Gal(L/F)
by restriction of automorphism from K to L.
Now we are led to the basic problem of determinig all Galois extensions K of a
given field F.
An element x of K is algebraic over a subfield F provided P(x) = 0 for some
nonzero polynomial P(X) in F[X]. Associated to x over F is a (unique) minimal
polynomial which has minimal degree and leading coefficient one among the poly-
nomials P = 0 with P(x) = 0. An extension K over F is algebraic provided every
x in K is algebraic. Every finite extension K over F is algebraic.
(1.4) Definition. An extension K over F is normal provided it is algebraic and for
every x in K the minimal polynomial of x over F has all its roots in K.
A finite extension K over F is normal if and only if K is generated by the roots
of a polynomial with coefficients in F. Every Galois extension K of F is seen to
be normal, because for x in K the conjugates s(x), where s ∈ Gal(K/F), are finite

in number and the minimal polynomial for x divided Q(X) = (X − s(x)). Note
Q(X) has coefficients in F since they are invariant under Gal(K/F).
(1.5) Definitions. A polynomial P(X) over a field F is called separable provided
its irreducible factors do not have repeated roots, An element x in an extension field
K over F is separable provided it is the root of a separable polynomial over F.An
extension K over F is separable provided every element of K is separable over F.

With our definition a separable extension is an algebraic extension. In character-
istic zero every algebraic extension is separable.

(1.6) Remark. An algebraic field extension is Galois if and only if it is normal and
separable. Over a field F, which is of characteristic zero or is finite, an algebraic
extension is Galois if and only if it is normal.
(1.7) Definition. A field F is algebraically closed provided any algebraic element x
in an extension K of F is in F. An algebraic closure F of F is an algebraic extension
F which is algebraically closed.

The algebraic closure F over F is a normal extension which is a Galois extension
if F is either a field of characteristic zero or a finite field. The extension is usually
infinite but it is the union of all finite subextensions, and indeed all finite normal
subextensions, and is also the direct (inductive) limit