MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $k$ be a field and $f\in k[a,b]$ an irreducible two-variable polynomial, $B := k[a,b]/(f)$ and $C$ the integral closure of $B$ in its fraction field.

I call $f$ good if it is irreducible in the ring $k[[a,b]]$ of formal power series; equivalently (Nagata, Local Rings, p. 122, Ex. 1), if $C$ has exactly one prime ideal lying above $\mathfrak m = (a,b)B$ (hence, $f$ being good just means that the curve $B$ is analytically irreducible at the origin).

I'm looking for examples of "good" polynomials $f$ such that the residue field $L$ of $C$ is a proper extension of $k$.

I can show (at least if $k$ is infinite) that $[L:k] \cdot r = \mu(f)$, where $\mu(f)$ is the degree of the lowest-degree summand of $f$, and $r$ is the ramification index of $\mathfrak m$ in $C$. Hence, it is clear that $\mu(f)$ must be large enough if one wants interesting examples.

The only "generic" class of examples I could come up with is the one where $f$ is homogeneous: Then $f$ being irreducible implies $f$ is good, and $[L:k] = \deg f$.

share|cite|improve this question

this is the answer_bot. Love your question. I am sure that in the meantime you have moved on and are studying fully faithful exact functors of derived categories of coherent sheaves, but I am still going to answer this one. Yeah!

We can construct examples by starting with a normal affine algebraic curve C over k and a closed point c of C with any given residue field L. If L/k is finite separable, this is always possible even with C being geometrically irreducible and smooth over k. I just made some examples where k has characteristic p > 0 and L is k[x, y]/(x^p - a, y^p - b) which I think generalizes. So there are lot's of L that occur.

Anyway, we next choose a general projection C ---> A^2_k (with coordinates a, b) which maps our chosen point c to (0, 0). The image of C is V(f) for some irreducible f. Since by construction (this is where the "general" above comes in) there is only one point of C above (0, 0) you get an example of what you want.

You can do this explicitly because you can make explicit curves C and then explicitly project and compute the equation f by taking a resultant. Good luck!

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.