Let $\zeta$ be a primitive $10^{th}-$root of unity. We know that its minimal polynomial over $\mathbb{Q}$ is $$f(z)=z^{4}-z^{3}+z^{2}-z+1,$$ and it splits over $\mathbb{C}$ as $$f(z)=(z-\zeta)(z-\zeta_{1})(z-\zeta_{2})(z-\zeta_{3}),$$ where each $\zeta_{k}$ is another primitive $10^{th}-$root of unity. And we call these $\zeta_{k}'$ the conjuate of $\zeta$ over $\mathbb{C}$.
Now, denote $\mathbb{C}_{p}$ the completion of $\overline{\mathbb{Q}}_{p}$ with respect to the extension of the $p-$adic absolute value, and $\mathbb{C}_{p}$ is called the field of p-adic complex number. Again, this minimal polynomial over $\mathbb{Q}$, i.e. $f(z)$, can split over $\mathbb{C}_{p}$ because it is algebraically closed, and it will also result in four conjugates, namely, $\xi_{1},\cdots,\xi_{4}$.
My question is, is it possible to find the explicit form of these four conjugates? If not, is it possible to compute the $p-$adic norm of them? for example, what is $|\xi_{1}-1|_{p}$?
Thank you!
Best Answer
Ok, in a more-beginner-like context, which is certainly fair-enough for Math Stack Exchange! (Maybe even one of the best reasons to have this site! :)
So, yes, in principle, there is no mystery here, in the sense that long-known, relatively elementary (by some standard) methods answer the question. Which is not to say that everyone on the street (or in the hallways in math depts) could instantly explain it.
So: the primitive 10th roots of unity are zeros of $\Phi_{10}(x)$, the 10th cyclotomic polynomial, which is (provably) $(x^{10}-1)/\Phi_1(x)\Phi_2(x)\Phi_5(x)=x^4-x^3+x^2-x+1$. Then $\zeta-1$ for any of the zeros satisfies $x^4-...+(1-1+1-1+1)$, and the constant term is $\pm 1$. That is, while all of the $\zeta^\ell-1$ are certainly algebraic integers, their product is $1$, so they are all (global!) units. So they are units $p$-adically for all $p$, whether or not they are in $\mathbb Q_p$ or some finite extension.