Linear representation theory of groups of prime-cube order

The discussion here is specifically intended to cater to odd primes $$p$$ and excludes the case $$p = 2$$, which is somewhat different. For more on $$p = 2$$, see linear representation theory of groups of order 8.

Degrees of irreducible representations
Due to the fact that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, for groups of order $$p^3$$ for a fixed prime $$p$$, the nilpotency class determines the degrees of irreducible representations. (This also holds for $$p = 2$$, although we are not discussing that case.

Splitting field
Two important notes:


 * Odd-order and ambivalent implies trivial, so none of these groups are ambivalent groups, i.e., none of them have all their character real-valued. Therefore, none of them are rational groups or rational-representation groups. This is in sharp contrast to the $$p = 2$$, where three of the five groups of order $$p^3$$ are rational groups and two of them are rational-representation groups.
 * Odd-order p-group implies every irreducible representation has Schur index one: All the irreducible representations have Schur index one, so they can be realized over the field generated by the character values. Note that this fact requires both the odd order and the fact that it is a group of prime power order. For $$p = 2$$, there is an example -- faithful irreducible representation of quaternion group -- that has Schur index two. Also, there is a group of order 63 that has an irreducible representation of Schur index three.

With these facts in mind, we present the list of minimal splitting fields. We denote by $$\zeta_n$$ a primitive $$n^{th}$$ root of unity. Note that the degree of the extension $$\mathbb{Q}(\zeta_n)$$ over $$\mathbb{Q}$$ is the Euler totient function $$\varphi(n)$$.