We are given a finite group H, an automorphism \(\tau \) of H of order r, a Galois extension L/K of fields of characteristic zero with cyclic Galois group \(\langle \sigma \rangle \) of order r, and an absolutely irreducible representation \(\rho :H\rightarrow \textsf {GL} (n,L)\) such that the action of \(\tau \) on the character of \(\rho \) is the same as the action of \(\sigma \). Then the following are equivalent.

\(\bullet \) \(\rho \) is equivalent to a representation \(\rho ':H\rightarrow \textsf {GL} (n,L)\) such that the action of \(\sigma \) on the entries of the matrices corresponds to the action of \(\tau \) on H, and

\(\bullet \) the induced representation \(\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )\) has Schur index one; that is, it is similar to a representation over K.

As examples, we discuss a three dimensional irreducible representation of \(A_5\) over \(\mathbb {Q}[\sqrt{5}]\) and a four dimensional irreducible representation of the double cover of \(A_7\) over \(\mathbb {Q}[\sqrt{-7}]\).

给定一个有限群H 、 r阶H的自同构\(\tau \) 、特征零域的伽罗瓦扩展L / K以及 r阶循环伽罗瓦群 \(\langle \ sigma \rangle \)，以及绝对不可约的表示\(\rho :H\rightarrow \textsf {GL} (n,L)\)使得\(\tau \)对\(\rho \)的字符的作用是相同的作为\(\sigma \)的动作。那么下面的都是等价的。

\(\bullet \) \(\rho \)等价于表示\(\rho ':H\rightarrow \textsf {GL} (n,L)\)使得\(\sigma \)对矩阵的条目对应于\(\tau \)对H的作用，并且

\(\bullet \)诱导表示\(\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )\) 的Schur 索引为 1；也就是说，它类似于K上的表示。

作为例子，我们讨论\(\mathbb {Q}[\sqrt{5}]\​​)上\ (A_5\ )的三维不可约表示和\(A_7\)上双重覆盖的四维不可约表示\(\mathbb {Q}[\sqrt{-7}]\)。