Algebra Solutions: Generating sets of the integers as an additive group

Generating sets of the integers as an additive group

Find a set of generators for $\mathbb{Z}$ .

Every integer can be written as a finite sum of 1s, so we have $\mathbb{Z} = \langle 1 \rangle$ . In particular, $\mathbb{Z}$ is a cyclic group.
Also, suppose $a$ and $b$ are integers with $\mathsf{gcd}(a,b) = 1$ . Then there exist integers $x,y$ with $ax + by = 1$ , so we also have $\mathbb{Z} = \langle a, b \rangle$ .

Algebra Solutions

Generating sets of the integers as an additive group

No comments:

Post a Comment