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.
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 $v_\infty(\frac 1X)=1$.
Remark
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 $p(X)=X-a$.