Definition:Splitting Field
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Definition
Of a polynomial
Let $K$ be a field.
Let $f$ be a polynomial over $K$.
A splitting field of $f$ over $K$ is a field extension $L / K$ such that:
- $f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$
for some $k \in K$, $\alpha_1, \ldots, \alpha_n \in L$.
We say that $f$ splits over $L$.
Of a set of polynomials
Let $K$ be a field.
Let $\FF$ be a set of polynomials over $K$.
A splitting field of $\FF$ over $K$ is a field extension $L / K$ such that for any $f \in \FF$:
- $f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$
for some $k \in K$, $\alpha_1, \ldots, \alpha_n \in L$.
We say that $\FF$ splits over $L$.
Minimal Splitting Field
Of a polynomial
Let $K$ be a field.
Let $f$ be a polynomial over $K$.
Let $L/K$ be a field extension of $K$.
Then $L$ is a minimal splitting field for $f$ over $K$ if $L$ is a splitting field for $f$ and no field extension properly contained in $L$ has this property.
Of a set of polynomials
Let $K$ be a field.
Let $\FF$ be a set of polynomials over $K$.
Let $L / K$ be a field extension of $K$.
Then $L$ is a minimal splitting field for $\FF$ over $K$ if $L$ is a splitting field for $f$ and no field extension properly contained in $L$ has this property.
Also see
Sources
- 1944: Emil Artin and Arthur N. Milgram: Galois Theory (2nd ed.) $\text{II}$: Field Theory: $\text{D}$ Splitting Fields