Exercise 111.1.8. Let $\nu : k[x]\setminus \{ 0\} \to {\mathbf Z}$ be a map with the following properties: $\nu (fg) = \nu (f) + \nu (g)$ whenever $f$, $g$ not zero, and $\nu (f + g) \geq min(\nu (f), \nu (g))$ whenever $f$, $g$, $f + g$ are not zero, and $\nu (c) = 0$ for all $c\in k^*$.

Show that if $f$, $g$, and $f + g$ are nonzero and $\nu (f) \not= \nu (g)$ then we have equality $\nu (f + g) = min(\nu (f), \nu (g))$.

Show that if $f = \sum a_ i x^ i$, $f\not= 0$, then $\nu (f) \geq min(\{ i\nu (x)\} _{a_ i\not= 0})$. When does equality hold?

Show that if $\nu $ attains a negative value then $\nu (f) = -n \deg (f)$ for some $n\in {\mathbf N}$.

Suppose $\nu (x) \geq 0$. Show that $\{ f \mid f = 0, \ or\ \nu (f) > 0\} $ is a prime ideal of $k[x]$.

Describe all possible $\nu $.

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