Linear representation theory of dihedral group:D16

This article gives specific information, namely, linear representation theory, about a particular group, namely: dihedral group:D16.
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Summary

We shall use the dihedral group with the following presentation:

$\langle a,x\mid a^{8}=x^{2}=e,xax^{-1}=a^{-1}\rangle$.

Item	Value
degrees of irreducible representations over a splitting field	1,1,1,1,2,2,2 maximum: 2, lcm: 2, number: 7, sum of squares: 16
Schur index values of irreducible representations over a splitting field	1,1,1,1,1,1,1
smallest ring of realization (characteristic zero)	$\mathbb {Z} [{\sqrt {2}}]$ or $\mathbb {Z} [x]/(x^{2}-2)$ (not sure -- need to check!)
smallest field of realization (characteristic zero)	$\mathbb {Q} ({\sqrt {2}})$ or $\mathbb {Q} [x]/(x^{2}-2)$
condition for a field to be a splitting field	?
smallest size splitting field	?
degrees of irreducible representations over the rational numbers	?

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
dihedral group	degree $n=8$, order $2n=16$	linear representation theory of dihedral groups