nLab separable closure

Definition

Definition

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

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

We call $\alpha \in \bar K$ separable over $K$ if the irreducible polynomial $f^\alpha_K$ of $\alpha$ over $K$ is separable.

A subfield $K \subset L \subset \bar K$ is called separable over $K$ if each $\alpha \in L$ is separable over $K$.

Definition

Let $K$ be a field and $\bar K$ an algebraic closure of $K$. The separable closure $K_S$ of $K$ is defined by

$$K_S \simeq \{x \in \bar K \mid x \, \text{ is separable over }\, K\}.$$

Remark

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

From xyz it follows that the inclusion $K \subset K_S$ is Galois.

Definition

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

Related concepts

• algebraic closure

• geometric point

• field extension

• absolute Galois theory

References

• Wikipedia, Separable closure