Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\alpha$ be an algebraic number and let $S$ be the orbit of $\alpha$ under the action of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$.

Do we have that $\# S $ is bounded from above by the degree of the splitting field of $\alpha$?

share|cite|improve this question
up vote 4 down vote accepted

Yes. The orbit of $\alpha$ consists of the conjugates of $\alpha$, whose number is the degree of $\alpha$, which is a lower bound for the degree of the splitting field of (the minimal polynomial for) $\alpha$.

share|cite|improve this answer

Your Answer

 
discard

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.