How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This seems to be a nontrivial problem. Finding all valuations is equivalent to finding all prime ideals, since the archimedean valuations are easy to describe. Finding all prime ideals depends on the Galois group of the extension and would be easy if the extension were abelian, which it rarely is (there are some for $n = 4$). The problem of finding the Galois group of the normal closure of these fields is discussed by Jacobson and Velez, The Galois group of a radical extension of the rationals Manuscr. Math. 67 (1990), 271-284.

For detailed information on integral bases and the decomposition of primes, a good place to start is Ribenboim's book "Algebraic Numbers". What you should do in general is find the possible decomposition and ramification subgroups, and then draw conclusions about the splitting of primes. There are exercises like these in Marcus' Number Fields.

Tags:

Absolute Value

Algebraic Number Theory

Related

From baby Hartshorne: showing the exterior of a circle is segment connected proof of inequality $e^x\le x+e^{x^2}$ How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$? how does expectation maximization work? Why is the 2nd derivative written as $\frac{\mathrm d^2y}{\mathrm dx^2}$? Erroneous numerical approximations of $\zeta\left(\frac{1}{2}\right)$? multiplication equivalent of the summation symbol Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ A measurable function on an atom is almost everywhere constant Finding the norm in the cyclotomic field $\mathbb{Q}(e^{2\pi i / 5})$ The $n$-disk $D^n$ quotiented by its boundary $S^{n-1}$ gives $S^n$ Distinct Sylow $p$-subgroups intersect only at the identity, which somehow follows from Lagrange's Theorem. Why?

Recent Posts