Let be a fixed algebraic closure of . If is any collection of non-zero polynomials, the splitting field of over is the subfield of generated by and the zeros of the polynomials in .
We call separable if it has no multiple zero in .
We call separable over if the irreducible polynomial of over is separable.
A subfield is called separable over if each is separable over .
Let be a field and an algebraic closure of . The separable closure of is defined by
We have that is a subfield of and that precisely if is a perfect field, in particular if the characteristic of is 0.
From xyz it follows that the inclusion is Galois.
The Galois group is called the absolute Galois group of .
Last revised on September 30, 2018 at 14:08:17. See the history of this page for a list of all contributions to it.