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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 $\mathbb{Q}$.2011-02-10
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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.2011-02-10
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    And you may reference the book *Algebraic number theory* by **Jurgen Neukirch**.2011-03-06
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    These Articles [A](http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/ostrowski.pdf), [B](http://www.ma.utexas.edu/users/voloch/Papers/vaalerch3.pdf), [C](http://www.bprim.org/cyclotomicfieldbook/bs2.pdf) and [D](http://www1.spms.ntu.edu.sg/~frederique/antchap6.pdf) may be of some use to you.2012-07-21

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