Generators of the multiplicative group of GF(2)

Question

GF(3) can be constructed as follows by polynomial $p(x)=x+1$:

$0=0$

$1=1$

$\alpha=2$

GF(5) can be constructed as follows by polynomial $p(x)=x+2$:

$0=0$

$1=1$

$\alpha=3$

$\alpha^2=\alpha\cdot\alpha=3\cdot3\bmod5=4$

$\alpha^3=\alpha^2\cdot\alpha=4\cdot3\bmod5=2$

Question is, what can be defined as element $\alpha$ in GF(2) and what primitive polynomial can be used to construct this field?

The possible confusion here is that the trivial group (with one element only) doesn't really need a generator. You may use $x-1$ as the generating polynomial in the same way as usual, but there really is nothing to generate :-) — Jyrki Lahtonen, Aug 17 '11 at 04:17

score 1 · Answer 1 · answered Aug 16 '11 at 21:09

1

Hint: How many element are there in the multiplicative group of $GF(2)$? Based on that can you point at a generator (if you need one)?

answered Aug 16 '11 at 21:09

Jyrki Lahtonen

133,153

2 elements, 0 and 1, but is it possible to define element $\alpha$ in GF(2)? – scdmb Aug 16 '11 at 21:11
@scdmb: Well, because $\alpha$ has to be in the multiplicative group, it cannot be zero. – Jyrki Lahtonen Aug 16 '11 at 21:22

score 1 · Accepted Answer · answered Aug 16 '11 at 21:15

1

A generator of the multiplicative group of a finite field is an element $\alpha$ such that the powers of $\alpha$ include all non-zero elements of the field. The multiplicative group of GF(2) has one element, and thus one generator: $\alpha = 1$.

answered Aug 16 '11 at 21:15

Ilmari Karonen

26,117

Generators of the multiplicative group of GF(2)

2 Answers2