I need some help to solve the following problem that appears on page 31 of the book of William Fulton entitled Algebraic Curves.

Exercise :

Let $ V = \mathbb{A}^1 $, $ \Gamma (V) = k[X] $, $ K = k(V) = k(X) $.

1) For each $ a \in k = V $, show that $ \mathcal{O}_a ( V ) $ is a discrete valuation ring with uniformizing parameter $ t = X-a $.

2) Show that $ \mathcal{O}_{\infty} = \{ \ \dfrac{F}{G} \in k(X) \mid \mathrm{deg} ( G ) \geq \mathrm{deg} (F) \ \} $ is also a discrete valuation ring, with uniformizing parameter $ t = \dfrac{1}{X} $.

Thanks in advance for your help.

  • $\begingroup$ Ok. So perhaps it is given $\;k\;$ is an algebraically closed field? $\endgroup$ – Timbuc Apr 5 '15 at 15:17

1) The discrete valuation at $a\in k$ of $\frac {c_r(X-a)^r+\dots+c_n (X-a)^n}{b_s(X-a)^s+\dots+b_m(X-a)^m} \; (r\leq n, s\leq m; c_r,b_s\neq 0)$ is $r-s$.
In particular $v_a(X-a)=1$.

2) The discrete valuation at $\infty$ of $\frac {c_0 +\dots+c_n X^n}{b_0+\dots+b_mX^m} \; (c_n,b_m\neq 0)$ is $m-n$.
In particular $v_\infty(\frac 1X)=1$.

Algebraic closedness of $k$ is irrelevant in the above.
However if $k$ is not algebraically closed there are more discrete valuations than those I have described, namely the $v_{p(X)}$ corresponding to monic irreducible polynomials $p(X)\in k[X]$ of degree $\gt1$.
Those new valuations are described by a formula similar to that in 1), which corresponds to the case $p(X)=X-a$.

  • $\begingroup$ In my book. We need to show that there is an irreducible element $ t \in R $ such that every non zero : $ z \in R $ may bee written uniquely in the form $ z = ut^n $ and $u$ a unit in $R$, and $n$ a nonnegative integer. How to show that ? $\endgroup$ – Bryan261 Apr 5 '15 at 17:00
  • $\begingroup$ In 1) $t$ is $X-a$ and in 2) $t$ is $\frac 1X$. Since $n$ is the valuation (respectively $r-s$ and $m-n$) you should be able to extract $u$ from the very explicit formula in the answer. Or just write brutally $u=\frac {z}{t^n}$ since you know $z,t$ and $n$ ...Also: beware that there is a clash in notation between your $n$ and my $n$ :-) $\endgroup$ – Georges Elencwajg Apr 5 '15 at 17:13
  • $\begingroup$ You are welcome, Bryan. $\endgroup$ – Georges Elencwajg Apr 5 '15 at 17:57

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