Let $$K$$ be a cyclic number field over $$\mathbb{Q}$$ of degree $$q=7$$ with Galois group $$G$$ and consider the $$\mathbb{F}_2$$ algebra over $$G$$, also known as the group ring. This is isomorphic to $$\mathbb{F}_2[x]/(x^q+1)$$. In the case $$q=7$$, the ideal $$(x^q+1)$$ splits mod $$2$$ as $$(x-1)(x^4+x^2+x+1)(x^4+x^2+x)$$. So even though the Galois group is cyclic, the $$\mathbb{F}_2$$ algebra over it is not.

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Then any Galois action in this $$\mathbb{F}_2$$ algebra can be taken to have a component in each of these split parts. We ignore $$(x-1)$$ for reasons I need to articulate better, it's somehow 'already handled' by cyclotomic context in my case, but... I am told the means to represent the Galois action in this way is to develop idempotents corresponding to the particular components and then take it as a linear combination of these idempotents. That is feasible and not a problem so far as ...

... calculating idempotents. One merely CRT's the thing and they emerge. The only problem is, they are $$x$$ and $$x+x^2$$, so I'm then not sure how to construct something out of those, because I missed the clue as to what scalars the 'linear combination' was using. I need to decompose $$x+x^3+x^4$$. This should be simple but... see missing clue noted above.

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