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

My question is whether for every extension of number fields $L\subset K$, and for every $f_0(x),...,f_n(x)$ in $K[x]$, there is some $\alpha\in L$ such that $$f_n(\alpha)T^n+...+f_1(\alpha)T+f_0(\alpha)$$ is irreducible as a polynomial in $K[T]$.

If $L=K$ this is known from Hilbert's Irreducibility Theorem. I find it hard to believe that there is a counter-example to this, but on the other hand I can't seem to conjure up a proof.

share|cite|improve this question
up vote 3 down vote accepted

The answer is yes, assuming that the two-variable polynomial $f_n(x)T^n + \dots + f_1(x)T + f_0(x)$ is irreducible over $K$.

This follows from the version of Hilbert's irreducibility theorem for number fields proved as Theorem 46 of p.298 of Schinzel's book Polynomials with special regard to reducibility: the relevant passage can be viewed on Google Books's%20irreducibility%20theorem&pg=PA298#v=onepage&q&f=false

In fact, if I'm reading it correctly, it looks like one has irreducibility for all rational integers $\alpha$ belonging to an appropriate residue class.

share|cite|improve this answer
Yes, you're right. Thank you. – Makhalan Duff Dec 3 '11 at 4:21
That extra assumption that you edited in is fine. What I had in mind is branched covers of curves where that condition is automatically satisfied. – Makhalan Duff Dec 3 '11 at 4:33

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.