Question

Let K be a number field of degree n with a fixed embedding in the complex numbers. Let | . | be the normalized absolute value given by that embedding. (The square of the ordinary absolute value if the embedding is non-real.) Does there exist a basis x_1 , ... , x_n for K over the rationals with the following property:

| r_1 x_1 | + ... + | r_n x_n | = the maximum of | r_1 x_1 + ... + r_n x_n |_v

where | . |_v runs through the normalized archimedean absolute values of K ?

Later: As posed, the answer is no. Suppose I started with the field generated by a cube root and was unlucky enough to start with the real embedding. Then I'd be in effect trying to show ( x + y + z )^2 > x^3 + y^3 + z^3 on the positive octant. But if I was lucky enough to start with the complex embedding ...

Answer 1

The archimedean absolute values are independent, so as posed, this question should fail as soon as your number field has at least two archimedean places.

Considering all places, both archimedean and finite, the only relation is that for x in K, the product of the v-adic absolute value of x taken over all places v is equal to one. Strong approximation will show that you can't expect any more non-trivial relationships between absolute values.