Galois Theory – Irreducible Components of a Cyclic Extension over $ \mathbb{Q} $

Question

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gal{Gal}$Let $ L $ be a cyclic Galois extension of $ \mathbb{Q} $ of degree $ 6 $. So $ G = \Gal(L/\mathbb{Q}) $ is a cyclic group of order $ 6 $. Then we have a homomorphism $ \phi : G \rightarrow \GL (L) $ defined by $ \phi(\sigma)(g) = \sigma(g) $ for all $ \sigma \in G $. Thus by representation theory we can consider $ L $ as a $\mathbb{Q}G $ module. Then $ L $ can be written as a direct-sum decomposition of $ r $ distinct irreducible submodules. What is the value of $ r $? And what are the irreducible components?

Answer 1

By the normal basis theorem, $L$ is the regular representation of $G$, i.e., $r = 6$ and the components are the $6$ distinct characters of $G$.

EDIT: As pointed out in the comments, particularly by @FrançoisBrunault, that is $L \otimes_{\mathbb Q} \mathbb C$ as a $\mathbb C[G]$-module; $L$ itself as a $\mathbb Q[G]$-module has $4$ summands, corresponding to the trivial and sign characters, and the two other pairs of complex conjugate characters.