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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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    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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