# Infinite places of a global function field

Let $$K$$ be a global function field, that is, a finite extension of $$\mathbb{F}_q(t)$$, where $$\mathbb{F}_q$$ is the finite field with $$q$$ elements.

What are the infinite places of $$K$$?

I know that $$\mathbb{F}_q(t)$$ has precisely one infinite place, namely that which corresponds to the change of variable $$t\mapsto 1/t$$. Furthermore, if $$K/\mathbb{F}_q(t)$$ is separable, then we can describe the infinite places of $$K$$ via the factorisation of the separable polynomial generating the extension, as in Corollary 13.6.

What happens when $$K/\mathbb{F}_q(t)$$ is not separable?

There are no infinite places in function fields, you mean places above $$t=\infty$$ ie. above $$u=t^{-1}=0$$.

$$K = F_q(u)(c_1,\ldots,c_n)$$, if $$g\in F_q(u)[x]$$ is irreducible and non-separable then $$g' = 0$$ so $$g = G(x^p)$$,

Let $$p^{e_j}$$ be $$c_j$$'s multiplicity in its minimal polynomial, then $$c_j^{p^{e_j}}$$ is separable over $$F_q(u)$$.

Thus with $$E = F_q(u)(c_1^{p^{e_1}},\ldots,c_n^{p^{e_n}})$$ we have $$K/E/F_q(u)$$ with $$K/E$$ purely inseparable and $$E/F_q(u)$$ separable, with the primitive element theorem $$E = F_q(u)(a)$$, let $$d$$ be the leading coefficient of $$a$$'s minimal polynomial $$\in F_q[u][x]$$, then $$E = F_q(u)(da)$$ where $$da$$'s minimal polynomial is $$f(u,x)\in F_q[u][x]_{monic}$$.

Since $$K^{p^{[K:E]}}\subset E$$, the places of $$K$$ above $$u=0$$ correspond to the places of $$E$$ above $$u=0$$ which correspond to the irreducible factors $$f_i(x)$$ of $$f(0,x)\in F_q[x]$$, each of them giving a maximal ideal $$\mathfrak{m}_i=(u,f_i(x))$$ of $$R=F_q[u][x]/(f(u,x))$$ which gives a principal maximal ideal $$(\pi_i)$$ of $$R_i=(R-\mathfrak{m}_i)^{-1} R$$ so that $$v_i(y) = k$$ if $$y\in (\pi_i^k),\not \in (\pi_i^{k+1})$$ is a discrete valuation on $$R_i$$ which gives a discrete valuation on $$E$$ and $$v_i(y) = p^{-[K:E]}v(y^{p^{[K:E]}})$$ is our discrete valuation on $$K$$.

• When you write $g=G(x^p)$, what do you mean? Tangentially, it seems strange that $t=\infty$ isn't called an infinite place. I know that it is not archimedean, but why must infinite and archimedean be synonymous when there is this other notion which needs to be distinguished...?
– user180579
Commented Oct 18, 2019 at 3:03