My question is the following: Are there algebraic norms on the fields $\mathbb{R}, \mathbf{Q_p}$ ''other'' than the absolute value, respectively $|\cdot|_p$?
Now phrasing more precisely: If generally $F$ is a field, an algebraic norm is a map $|\cdot| : K \to [0, \infty)$ such that
1) $|x| = 0 \iff x = 0$
2) $|xy| = |x||y| \forall x,y \in F$
3) $|x + y| \leq |x| + |y| \forall x,y \in F$
Two such norms $|\cdot|, ||\cdot||$ are called equivalent if and only if they generate the same topology. One can show that this is the case iff. there is an $s > 0$ such that $|x| = ||x||^s$ for all $x$. The question is: On the fields $\mathbb{R}, \mathbf{Q_p}$ where the latter one means the p-adic numbers, are there algebraic norms that are not equivalent to the usual absolute value, respectively the p-adic norm?
Of course, every such norm induces a norm on $\mathbb{Q}$, so restricted to $\mathbb{Q}$ it must be either trivial or equivalent to one of the norms mentioned above by the Thm. of Ostrowski. The problem is that i was unable to extend that to the whole field.
Thanks in advance
Fabian Werner