Definition. A function $\nu$ defined in a field $K$ is called an exponent valuation or shortly an exponent of the field, if it satisfies the following conditions:

1. $\nu(0) = \infty$ and $\nu(\alpha)$ runs all rational integers when $\alpha$ runs the non-zero elements of $K$ .
2. $\nu(\alpha\beta) = \nu(\alpha)+\nu(\beta)$ .
3. $\nu(\alpha+\beta) \geqq \min\{\nu(\alpha),\,\nu(\beta)\}$ .
   

Properties.
$\nu(1) = 0$
$\nu(-\alpha) = \nu(\alpha)$
$\displaystyle\nu\left(\frac{\alpha}{\beta}\right) = \nu(\alpha)\!-\!\nu(\beta)$
$\nu(\alpha^n) = n\,\nu(\alpha)$
$\nu(\alpha_1+\ldots+\alpha_n) \geqq \min\{\nu(\alpha),\,\ldots,\,\nu(\alpha_n)\}$
$\nu(\alpha+\beta) = \min\{\nu(\alpha),\,\nu(\beta)\} \quad \mbox{if}\;\;\;\nu(\alpha) \neq \nu(\beta)$

Example. If an integral domain $\mathcal{O}$ has a divisor theory $\mathcal{O}^* \to \mathfrak{D}$ , then for each prime divisor $\mathfrak{p}$ there is an exponent valuation $\nu_{\mathfrak{p}}$ of the quotient field $K$ of $\mathcal{O}$ . It is given by

Different prime divisors $\mathfrak{p}$ and $\mathfrak{q}$ determine different exponents $\nu_{\mathfrak{p}}$ and $\nu_{\mathfrak{q}}$ , since the condition 3 of the definition of divisor theory guarantees such an element $\gamma$ of $\mathcal{O}$ which in divisible by $\mathfrak{p}$ but not by $\mathfrak{q}$ ; then $\nu_{\mathfrak{p}}(\gamma) \geqq 1$ , $\nu_{\mathfrak{q}}(\gamma) = 0$ .

Theorem. Let $\nu_1,\,\ldots,\,\nu_r$ be different exponents of a field $K$ . Then for arbitrary set $n_1,\,\ldots,\,n_r$ of integers, there exists in $K$ an element $\xi$ such that $$\nu_1(\xi) = n_1,\;\;\ldots,\;\;\nu_r(\xi) = n_r.$$

The proof of this theorem is found in [1].

Bibliography

1

S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).