Timeline for Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?
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| Apr 8 at 10:07 | comment | added | Thrash | Update: If there is more than one such element, we can consider all such elements together with $e$, which form a subgroup that (I think) is isomorphic to a $k$-fold copy of the cyclic group of order $2$ (where $k>1$), and it is not difficult to see that the product of all these elements yields $e$ because in each component, the element of order $2$ occurs an even number of times. | |
| Apr 7 at 22:10 | comment | added | Thrash | What if there is more than one element of order 2? Then we just now that the product of all elements is the product of the elements of order 2 (which could be different from $e$). | |
| Jun 20, 2022 at 5:26 | comment | added | KConrad | For prime $p$, $-1 \bmod p^\alpha$ has order $2$ unless $p^\alpha = 2$. So by the Chinese rem. thm, there is more than one unit mod $n$ of order $2$ if $n$ has more than one odd prime factor or if $n$ has one odd prime factor and is divisible by $4$. There are three elements of order $2$ in the units mod $2^\alpha$ if $\alpha \geq 3$. The only moduli $n > 1$ left for which there could be a unique unit of order $2$ are $2$, $4$, $p^\alpha$, and $2p^\alpha$ for odd primes $p$. For these moduli except $2$, only one unit has order $2$. We don’t need to know the moduli whose units are cyclic. | |
| Jun 20, 2022 at 1:55 | history | edited | Ofir Gorodetsky | CC BY-SA 4.0 |
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| Jun 20, 2022 at 0:37 | history | edited | Ofir Gorodetsky | CC BY-SA 4.0 |
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| Jun 19, 2022 at 17:15 | history | answered | Ofir Gorodetsky | CC BY-SA 4.0 |