# Splitting field

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This term associates to every group, a corresponding field property. In other words, given a field, every field either has the property with respect to that group or does not have the property with respect to that group

## Definition

### In terms of realization of irreducible representations

A splitting field for a finite group is a field satisfying both the following two conditions:

1. Every linear representation of the group is completely reducible. This is equivalent to saying that the characteristic of the field does not divide the order of the group.
2. Any linear representation of the group over any extension of the given field, is equivalent to a linear representation over the field itself

### In terms of the character ring

A splitting field for a finite group is a field satisfying both the following two conditions:

1. Every linear representation of the group is completely reducible. This is equivalent to saying that the characteristic of the field does not divide the order of the group.
2. The character ring of the group over the field is equal to the character ring of the group over any extension of the field. Here, the character ring of a group over a field is the ring of $\Z$-linear combinations of characters of representations of the group realizable over that field.

Note that in some alternative definitions, only condition (2) is imposed for being a splitting field, thus also including the modular case where not all representations are completely reducible.

## Examples

Group Order Smallest splitting field in characteristic zero Necessary and sufficient condition for splitting field in characteristic coprime to group order Condition on $q$ for field of size $q$, $q$ coprime to group order
trivial group 1 field of rational numbers any field any $q$
cyclic group:Z2 2 field of rational numbers any field any $q$
cyclic group:Z3 3 $\mathbb{Q}[x]/(x^2 + x + 1)$ -- need to adjoin primitive cuberoot of unity any field where $x^2 + x + 1$ splits $3$ divides $q - 1$
cyclic group:Z4 4 $\mathbb{Q}[x]/(x^2 + 1)$ -- need to adjoin squareroot of $-1$ any field where $x^2 + 1$ splits $4$ divides $q - 1$
Klein four-group 4 field of rational numbers any field any $q$
cyclic group:Z5 5 $\mathbb{Q}[x]/(x^4 + x^3 + x^2 + x + 1)$ any field where $x^4 + x^3 + x^2 + x + 1$ splits, or has a root $5$ divides $q - 1$
symmetric group:S3 6 $\mathbb{Q}$ any field any $q$
cyclic group:Z6 6 $\mathbb{Q}[x]/(x^2 + x + 1)$ -- need to adjoin primitive cuberoot of unity any field where $x^2 + x + 1$ splits $3$ divides $q - 1$
group of prime order $p$ $p$ $\mathbb{Q}[x]/(x^{p-1} + \dots + x + 1)$ any field where $x^p - 1$ splits, equivalently, $x^{p-1} + \dots + x + 1$ has a root $p$ divides $q - 1$
cyclic group of order $n$ $n$ $\mathbb{Q}[x]/\Phi_n(x)$, where $\Phi_n$ is the cyclotomic polynomial any field where $x^n - 1$ splits $n$ divides $q - 1$
dihedral group:D8 8 field of rational numbers any field any $q$