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I have a homework problem that uses the term: "a discrete normed non-trivial valuation" on a field.

We've defined the discrete trivial valuation in class, so that part is clear.

I think the natural meaning of normed is that the discrete valuation takes only positive values. I just want to make sure though.

Is this correct?

Additionally, is $v_{\infty}$ notation for the discrete valuation that maps every element to $\infty$?

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No, $v_\infty$ is the valuation at the "infinite" place, which is just a fancy name for the standard archimedean valuation. –  Zhen Lin Mar 16 '12 at 22:15
thanks very much! –  Kyle Schlitt Mar 16 '12 at 22:33
I think "normed" here means the valuation of a uniformizing element is $1$. –  user18119 Mar 16 '12 at 22:56
We spoke about normalizing/standardizing discrete valuations. So that is surely what this means. Thanks for jogging my memory! –  Kyle Schlitt Mar 16 '12 at 23:12

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