If $p$ a prime number $\mathbb{Z}^{\times}_p$ is always isomorphic to $(\mathbb{Z}_{p-1},+)$

Asked Nov 16 '17 at 12:30

Active Nov 16 '17 at 13:09

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Here https://math.stackexchange.com/a/1127555/283262

someone stated: "If $p$ a prime number $\mathbb{Z}^{\times}_p$ is always isomorphic to $(\mathbb{Z}_{p-1},+)$" What is this Theorem called? Where can i find it? How to proove it?

Thank you in advance.

edited Nov 16 '17 at 13:09

asked Nov 16 '17 at 12:30

thetha

3

You can find it in every introductory Number Theory textbook ever published. – Gerry Myerson Nov 16 '17 at 12:32
The multiplicative group of a finite field is cyclic. – egreg Nov 16 '17 at 12:38
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Here is a post asking a slightly more general question. – Arthur Nov 16 '17 at 12:38
@Arthur Thank you, this is very intresting – thetha Nov 16 '17 at 13:05
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@Vinyl_coat_jawa That comment makes it look much easier than it really is. It is nowhere near sufficient to note that they have the same order. – Tobias Kildetoft Nov 16 '17 at 15:00
@TobiasKildetoft true, but still it was a hint I think. it is indeed not sufficient – Vinyl_cape_jawa Nov 16 '17 at 15:09
@TobiasKildetoft but you are right, it may be misleading. I will remove it :) – Vinyl_cape_jawa Nov 16 '17 at 15:24
dont worry, im ok :) – thetha Nov 16 '17 at 19:14

If $p$ a prime number $\mathbb{Z}^{\times}_p$ is always isomorphic to $(\mathbb{Z}_{p-1},+)$

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