Let $K$ be a Galois extension of $\mathbb{Q}$ of degree $n$. Pick some primitive element and take the roots $a\_1, ..., a\_n$ of its minimal polynomial. Then the evaluation map $\mathbb{Q}[x\_1, ..., x\_n] \to K$ with $x\_i \mapsto a\_i$ gives a surjection to $K$. The kernel is a maximal ideal $M$ of algebraic relations between $a\_1, ..., a\_n$. Is it true that $M$ is generated as an ideal by the polynomials of $\mathbb{Q}[x\_1, ..., x\_n]$ that are fixed by the Galois group of $K$ acting on $\mathbb{Q}[x\_1, ..., x\_n]$ and are contained within $M$?