# Bound for the degree of the field of definition for a closed point of a variety

While attempting to prove some existence theorem for matrices over $\mathbb{F}_{2^k}$ I've come across the following problem concerning fields of definition for closed point of, say affine, varieties.

Let $X/k$ be a variety defined over a field $k$ given by a $k$-algebra $A = k[x_1,\ldots,x_n]/(f_1,..,f_i)$ such that $X = \textrm{Spec}(A)$. Is there any upper bound for the minimal degree $[l:k]$ for a closed point $\textrm{Spec}(l) \to X$, that depends on the field $k$, the degrees of $f_i$'s and the parameter $n$. For example, can we say something for $k$ a $C_r$ field.

In particular, are you familiar with any reference concerning the special case of finite fields or other $C_1$ fields.

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The Lang-Weil (Am J Math 1954) bound gives such a result for finite fields. – Felipe Voloch Apr 22 '13 at 20:25
There is something wrong with your TeX. The math doesn't show properly: specifically, the math italic alphabetic characters don't show at all. – Laurent Moret-Bailly Apr 23 '13 at 7:03
Thank you Felipe. I have also found this lecture notes math.lsa.umich.edu/~mmustata/lecture7.pdf , that describe Lang-Weil's result in more modern language. Laurent, unfortunately I can't experience anything in my web browser. Can I just use dollars \\$ to introduce TeX formulas?? – Maciekrt Apr 23 '13 at 9:55