I am following John Milne's notes on algebraic number theory and i'm having a hard time understanding his proof of the uniqueness of the extension of an absolute value to a finite separable extension L of K. To make things short, we end up with a ring A[b] (where A corresponds to the x lying in K with norm <= 1 and we will call p the unique maximal ideal of A) which is such that: On one hand, the quotient A[b]/pA[b] is a local ring. On the other hand, A[b] has 2 prime ideals containing p. How is that a contradiction? Any other sketch of proof is welcome though. Thank you for your time
2026-03-28 14:37:46.1774708666
Extending nonarchimedian absolue value over a discrete complete valuation field
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