[Math] Galois extension definition.

abstract-algebradefinitiongalois-theoryterminology

Let $L,K$ be fields with $L/K$ a field extension. We say $L/K$ is a Galois extension if $L/K$ is normal and separable.

I don't fully understand this definition, is it saying that

1) $L$ has to be the splitting field for some polynomial in $K[x]$ and that polynomial must not have any repeated roots, or is it saying that

2) $L$ has to be the splitting field for all polynomials in $K[x]$ and all polynomials must not have repeated roots?

Best Answer

We define a Galois extension $L/K$ to be an extension of fields that is

  1. Normal: if $x\in L$ has minimal polynomial $f(X) \in K[X]$, and $y$ is another root of $f$, then $y\in L$.
  2. Separable: if $x\in L$ has minimal polynomial $f(X) \in K[X]$, then $f$ has distinct roots in its splitting field.

When $L/K$ is a finite extension, these conditions are equivalent to $L$ being the splitting field of a separable polynomial $f(X) \in K[X]$ - i.e. your condition $1$. This is a fact which is proven in any course in Galois theory. See for example Theorem 3.10 in these lecture notes.

Your condition $2$ is certainly false: for example $\mathbb Q(\sqrt2)/\mathbb Q$ is a Galois extension, but is not the splitting field of $X^5+3X+2$ or of any other (irreducible) polynomial other than $X^2-2$.

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