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consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are these finitely many? In general, if $K$ is a number field how to find them? Thanks

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The absolute values behave much like prime ideals of the ring of integers of $K$. Each prime $p$ gives rise to an absolute value on $\mathbb{Q}$, and the primes of $\mathcal{O}_K$ lying over $(p)$ give rise to absolute values on $K$ that are closely related to the $p$-adic value in $\mathbb{Q}$. The different embeddings of $K$ into $\mathbb{C}$ give rise to absolute values that are related to the usual absolute value of $\mathbb{Q}$. –  Arturo Magidin Feb 10 '11 at 19:16
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In particular, yes, there are only finitely many absolute values of $K$ that extend absolute values on $\mathbb{Q}$, up to equivalence. –  Arturo Magidin Feb 10 '11 at 19:32
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And you may reference the book Algebraic number theory by Jurgen Neukirch. –  awllower Mar 6 '11 at 3:22
    
These Articles A, B, C and D may be of some use to you. –  Iyengar Jul 21 '12 at 8:55

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