Editing Zech's logarithm (section)

==Examples==

Let {{nowrap|''α'' ∈ GF(2<sup>3</sup>)}} be a [[root of a polynomial|root]] of the [[primitive polynomial (field theory)|primitive polynomial]] {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> + 1}}.  The traditional representation of elements of this field is as [[polynomial]]s in α of [[degree of a polynomial|degree]] 2 or less.

A table of Zech logarithms for this field are {{nowrap|1=''Z''(−∞) = 0}}, {{nowrap|1=''Z''(0) = −∞}}, {{nowrap|1=''Z''(1) = 5}}, {{nowrap|1=''Z''(2) = 3}}, {{nowrap|1=''Z''(3) = 2}}, {{nowrap|1=''Z''(4) = 6}}, {{nowrap|1=''Z''(5) = 1}}, and {{nowrap|1=''Z''(6) = 4}}. The multiplicative order of ''α'' is 7, so the exponential representation works with integers modulo 7.

Since ''α'' is a root of {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> + 1}} then that means {{nowrap|1=''α''<sup>3</sup> + ''α''<sup>2</sup> + 1 = 0}}, or if we recall that since all coefficients are in&nbsp;GF(2), subtraction is the same as addition, we obtain {{nowrap|1=''α''<sup>3</sup> = ''α''<sup>2</sup> + 1}}.

The conversion from exponential to polynomial representations is given by
:<math>\alpha^3 = \alpha^2 + 1</math> (as shown above)
:<math>\alpha^4 = \alpha^3 \alpha = (\alpha^2 + 1)\alpha = \alpha^3 + \alpha = \alpha^2 + \alpha + 1</math>
:<math>\alpha^5 = \alpha^4 \alpha = (\alpha^2 + \alpha + 1)\alpha = \alpha^3 + \alpha^2 + \alpha = \alpha^2 + 1 + \alpha^2 + \alpha = \alpha + 1</math>
:<math>\alpha^6 = \alpha^5 \alpha = (\alpha + 1)\alpha = \alpha^2 + \alpha</math>

Using Zech logarithms to compute ''α''<sup>{{hairsp}}6</sup> + ''α''<sup>{{hairsp}}3</sup>:
:<math>\alpha^6 + \alpha^3 = \alpha^{6 + Z(-3)} = \alpha^{6 + Z(4)} = \alpha^{6 + 6} = \alpha^{12} = \alpha^5,</math>
or, more efficiently,
:<math>\alpha^6 + \alpha^3 = \alpha^{3 + Z(3)} = \alpha^{3 + 2} = \alpha^5,</math>
and verifying it in the polynomial representation:
:<math>\alpha^6 + \alpha^3 = (\alpha^2 + \alpha) + (\alpha^2 + 1) = \alpha + 1 = \alpha^5.</math>