P-adic valuation

Fix a prime number $p$. The $p$-adic valuation (or order) $\nu_p({\cdot})$ on the field of rational numbers is defined by $\nu(a/b) = r$ where $a,b$ are integers and $a/b = p^r \cdot a'/b'$ with $a',b'$ coprime to $p$; set $\nu_p(0) = \infty$. The $p$-adic norm $\Vert{\cdot}\Vert_p$ is defined by $\Vert x \Vert_p = p^{-\nu_p(x)}$, with $\Vert 0 \Vert_p = 0$.

The $p$-adic order satisfies

• $\nu_p(x y) = \nu_p(x) + \nu_p(y)$;
• $\nu_p(x+y) \ge \min\{\nu_p(x),\nu_p(y)\}$, with equality if the two terms are unequal.

The $p$-adic norm correspondingly satisfies

• $\Vert xy \Vert_p = \Vert x \Vert_p \Vert y \Vert_p$;
• $\Vert x+y \Vert_p \le \max\{\Vert x \Vert_p, \Vert y \Vert_p\}$.

The $p$-adic norm is thus an ultrametric norm. The $p$-adic numbers are the completion of the rationals with respect to the $p$-adic norm, and the $p$-adic valuation extends to the field of $p$-adic numbers.

Ostrowki's theorem states that the only norms on the field of rational numbers are the $p$-adic norms and the usual absolute value.

References

• Cassels, J.W.S. "Local fields" London Mathematical Society Student Texts 3 Cambridge University Press (1986) ISBN 0-521-31525-5 Zbl 0595.12006