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I am looking at the first two examples in Paul Garretts notes on p-adic number theory.

The first example is computing the Newton polygon of $x^5+2x^2+5$ over $\mathbb{Q}_2$. I think this is the lower convex hull of $(0,1),(1,0),(2,0.5),(3,0),(4,0),(5,1)$. So I thought the Newton polygon should just be a horizontal line segment going from $(0,1)$ to $(5,1)$. Of course, I am completely wrong, but I am not sure what is the source of error. Are the points whose convex hull I am supposed to consider correctly identified?

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up vote 2 down vote accepted

HINT $\ $ Below is the Newton polygon graphed by Filaseta's Java applet. His web pages have much of interest on Newton polygons. alt text

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OK, got it. I was considering the $y$ values to $p$-adic absolute values, when they should be just valuations. – Derek Scavo Dec 13 '10 at 7:06

You have miscalculated the $2$-adic orders of the coefficients, and probably confused the concept of $p$-adic order with something else. In particular, the $p$-adic order of a $p$-adic number is an integer, but your third point has $y$-coordinate 0.5. As a reminder: the $p$-adic order of an integer $x$ in $\mathbb{Z}_p$ is the highest integer $v$ such that $x\in {\mathfrak m}^v$, where ${\mathfrak{m}}=p\mathbb{Z}_p$ is the maximal ideal of $\mathbb{Z}_p$. E.g. the $2$-adic order of 5 is 0, since 5 is a 2-adic unit, so the first point of the Newton polygon has coords (0,0). The $p$-adic order of 0 is $\infty$ by convention. The $2$-adic order of 2 is 1, since 2 is a uniformiser. And so on.

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