I've thinking about the possibility how is the image of a two dimensional complex representations of absolute galois group of $\mathbb Q$, $G_\mathbb Q$.
I thought I had a proof about what the image is like but it was wrong.
I writing my attempts and conjectures about this.
Let
$$\rho:G_\mathbb Q\to\text{GL}_2(\mathbb C) $$
be a continuous unitary irreducible representation of $G_\mathbb Q$.
It is well known that $\rho$ factors through a finite Galois extension $K/\mathbb Q$ i.e. we have
$$\rho:\text{Gal}(K/\mathbb Q)\to\text{GL}_2(\mathbb C).$$
I have two question about the image of this representations:
- There exist an a representations equivalent to $\rho$ such that the matrix coefficients always are algebraic numbers i.e. if the image consists of matrix of the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ then, a,b,c,d are algebraic numbers?
- I've thinking in the possibility that the response of the question is affirmative, but I have not found the right way. And even, I think that if the conjecture is wrong then you can look at certain special elements such that the matrix coefficients are algebraic, for example on conjugacy class of the Frobenius elements.
I appreciate you any suggestion, or an counterexample.
Thanks you all
The number of irreducible representations of a finite group G over an algebraically closed field K of characteristic zero is equal to the number of conjugacy classes — this follows at once from Wedderburn's theorem applied to the group algebra KG. In particular, the numbers of irreducible representations of G over $\mathbb C$ and over the algebraic closure of $\mathbb Q$ are the same.
This implies immediately that every complex representation is defined over the algebraic numbers. Since it is moreover finite dimensional and the group has finitely many elements, it is defined over a finite extension of $\mathbb Q$ — consider the field generated by all entries of all matrices representing all elements of the group.