I came across an exercise in a book which I do not how to solve. Let $f(x) = x^5 -80x -2.$ Let $\alpha$ be a zero of this polynomial. Furthermore let $K = \mathbb{Q}(\alpha)$ be a number field. These are the questions that are asked:
1- Is this polynomial irreducible and why?
2- Find an integer $n$ such that $f(n) = 1.$
3- Show that $\alpha - n \in \mathcal O_K$
For part 1, For prime $p = 2$ we observe that $p$ does not divide the coefficient of the leading term but divides the other coefficients namely $-80, -2.$ Also $p^2 = 4$ does not divide $2.$ Therefore $f(x)$ by Eisenstein Irreducibility Criterion, is irreducible. Is this correct?
I am formally trained as a mechanical engineer and during this lockdown I am teching myself algebraic number theory. Please kindly provide as much details as possible as I am a bit clueless. Many thanks.
Your solution to 1) is correct. For 2), I can promise you that you will solve it with a few minutes of experimentation. Try things!
For 3), I think you have omitted the important piece of information that $\alpha$ is a root of $f$! Furthermore, the question as stated is trivial: $\alpha$ is the root of a monic integer polynomial, so it's an algebraic integer, and a regular integer $n$ is certainly an algebraic integer, and your book should have a proof that the sum of two algebraic integers is an algebraic integer.
I think the original problem was actually to show that $\alpha - n \in\mathcal{O}_K^\times$, i.e. that $\alpha-n$ is an algebraic integer whose inverse is also an algebraic integer.
My hint for showing this is to consider the following ratio: $$\frac{f(\alpha)-f(n)}{\alpha-n}$$
You will need the following fact, which holds for any integer polynomial $f$: $$\frac{f(X)-f(Y)}{X-Y}\in \mathbb{Z}[X,Y]$$
(to see this, look at each term of $f$)