Editing Zech's logarithm (section)

==Definition==
Given a [[Primitive element (finite field)|primitive element]] <math>\alpha</math> of a finite field, the Zech logarithm relative to the base <math>\alpha</math> is defined by the equation
:<math>\alpha^{Z_\alpha(n)} = 1 + \alpha^n,</math>
which is often rewritten as
:<math>Z_\alpha(n) = \log_\alpha(1 + \alpha^n).</math>
The choice of base <math>\alpha</math> is usually dropped from the notation when it is clear from the context.

To be more precise, <math>Z_\alpha</math> is a [[function (mathematics)|function]] on the [[integer]]s [[modular arithmetic|modulo]] the multiplicative order of <math>\alpha</math>, and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol <math>-\infty</math>, along with the definitions
:<math>\alpha^{-\infty} = 0</math>
:<math>n + (-\infty) = -\infty</math> 
:<math>Z_\alpha(-\infty) = 0</math>
:<math>Z_\alpha(e) = -\infty</math>
where <math>e</math> is an integer satisfying <math>\alpha^e = -1</math>, that is <math>e=0</math> for a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] 2, and <math>e = \frac{q-1}{2}</math> for a field of [[parity (mathematics)|odd]] characteristic with <math>q</math> elements.

Using the Zech logarithm, finite field arithmetic can be done in the exponential representation:
:<math>\alpha^m + \alpha^n = \alpha^m \cdot (1 + \alpha^{n-m}) = \alpha^m \cdot \alpha^{Z(n-m)} = \alpha^{m + Z(n-m)} </math>
:<math>-\alpha^n = (-1) \cdot \alpha^n = \alpha^e \cdot \alpha^n = \alpha^{e+n}</math>
:<math>\alpha^m - \alpha^n = \alpha^m + (-\alpha^n) = \alpha^{m + Z(e+n-m)} </math>
:<math>\alpha^m \cdot \alpha^n = \alpha^{m+n}</math>
:<math>\left( \alpha^m \right)^{-1} = \alpha^{-m}</math>
:<math>\alpha^m / \alpha^n = \alpha^m \cdot \left( \alpha^n \right)^{-1} = \alpha^{m - n}</math>
These formulas remain true with our conventions with the symbol <math>-\infty</math>, with the caveat that subtraction of <math>-\infty</math> is undefined. In particular, the addition and subtraction formulas need to treat <math>m = -\infty</math> as a special case.

This can be extended to arithmetic of the [[projective line]] by introducing another symbol <math>+\infty</math> satisfying <math>\alpha^{+\infty} = \infty</math> and other rules as appropriate.

For fields of characteristic 2,
:<math>Z_\alpha(n) = m \iff Z_\alpha(m) = n</math>.