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I had a slight confusion about Galois groups over a base field which is complete with respect to a discrete valuation.

We know that there are irreducible polynomials such as $X^3+X^2+2X-8$ where the roots have different valuation in the splitting field. In this example, take $K=\mathbb{Q}_2$ to be the base field and consider its splitting field. By Newton's Polygon, the three roots have valuations $0$, $1$ and $2$. (See the algebraic number theory notes at www.jmilne.org/math/)

However, it is also noted that Galois groups preserve the valuation regardless of what kind of extension it is (unramified, totally ramified, etc). This is by the uniqueness of extension of an absolute value in a complete field. (Analogous to the result, all norms of a finite real dimensional vector space are Lipschitz equivalent.)

To be more specific, if $v$ is a valuation then $v':x\mapsto v(\sigma(x))$ for $\sigma\in Gal(L/K)$ is another valuation and $Im(v)=Im(v')$, they must be the same valuation. (i.e. no nonsense about scale factors)

My question is that these two results seem to be contradicting each other. If we take $K=\mathbb{Q}_2$ and $L$ to be the splitting field of $X^3+X^2+2X-8$, then a field automorphism in the Galois group needs to be able to send an element $\alpha_1$, a root of $X^3+X^2+2X-8$ to another root $\alpha_2$ of a different valuation.

So where did I reason wrongly?

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Your problem is that you didn’t use the fact that Newton Polygon tells you that your cubic is by no means irreducible: in fact, it splits over $\Bbb Q_2$ into three linear factors. In other words, as a $\Bbb Q_2$ polynomial, its splitting field is $\Bbb Q_2$.

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  • $\begingroup$ Thank you, that makes sense. $\endgroup$ – daruma Dec 25 '18 at 5:29

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