I am currently studying Galois Theory and am having trouble understanding group notation.
What does $$\mathbb{Z}/n\mathbb{Z}$$
mean? I understand that its an additive group of modulo $n$ but what would the elements of $$\mathbb{Z}/6\mathbb{Z}$$ be for example?
Best Answer
$\mathbb{Z}/6\mathbb{Z}$ is a quotient group of $\mathbb{Z}$ by the (trivially normal) subgroup $6\mathbb{Z}$, and it's formal elements are 6 cosets: $\{ 0 + 6\mathbb{Z}, 1 + 6\mathbb{Z}, \dots, 5 + 6\mathbb{Z} \}$. These can be identified by chosing a single representative, like $0$ for $0 + 6\mathbb{Z}$ (sometimes denoted $\bar 0$ or $[0]$) and so on.
If you have trouble understanding this, I suggest you to study general group & ring theory first, before trying Galois theory, which is a bit more advanced.