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separable closure

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Definition

Definition

Let K¯\bar K be a fixed algebraic closure of KK. If FK[X]{0}F \subset K[X] - \{0\} is any collection of non-zero polynomials, the splitting field of FF over KK is the subfield of K¯\bar K generated by KK and the zeros of the polynomials in FF.

We call fK[X]{0}f \in K[X]- \{0\} separable if it has no multiple zero in K¯\bar K.

We call αK¯\alpha \in \bar K separable over KK if the irreducible polynomial f K αf^\alpha_K of α\alpha over KK is separable.

A subfield KLK¯K \subset L \subset \bar K is called separable over KK if each αL\alpha \in L is separable over KK.

Definition

Let KK be a field and K¯\bar K an algebraic closure of KK. The separable closure K SK_S of KK is defined by

K S{xK¯|xisseparableoverK}. K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\} \,.
Remark

We have that K SK_S is a subfield of K¯\bar K and that K SK¯K_S \simeq \bar K precisely if KK is a perfect field, in particular if the characteristic of KK is 0.

From xyz it follows that the inclusion KK SK \subset K_S is Galois.

Definition

The Galois group Gal(K S/K)Gal(K_S/K) is called the absolute Galois group of KK.

References

Last revised on September 30, 2018 at 10:08:17. See the history of this page for a list of all contributions to it.