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In my algebraic structures textbook there is a proof for the theorem if $p$ is a prime then $(p-1)!\equiv -1\pmod p\ $.

Proof: Since p is prime, each element $1,2,3...(p-1)$ in $\Bbb Z_p $ has an inverse, hence the pairs of inverses will cancel out leaving only the self-inverse elements 1 and -1 and thus $(p-1)! = 1(-1) = -1 $ in $\Bbb Z_p $.

My question is how is that possible when the elements of $(p-1)! $ are all positive? I must be confusing something...

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  • $\begingroup$ Do you know what the symbol $-1$ means in the context of $\mathbb{Z}_p$? It's something about divisibility, not negativity. $\endgroup$ – user296602 Jan 26 '16 at 2:54
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"Positive" and "negative" don't make sense as comparisons to zero in $\mathbb{Z}_p$ - after all, if you add $1$ to itself $p$ times you get zero. And if you do it $p - 1$ times, you get something congruent to $-1$ modulo $p$. The statement that $(p - 1)! \equiv -1 \pmod p$ means that

$$p \mid (p - 1)! + 1$$

as a statement about divisors. What the proof is doing is collecting a bunch of pairs that multiply to $1$ modulo $p$ - that is, if you divide the product by $p$ you get a remainder of $1$.

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$$(p-1)!\equiv -1\pmod p\ $$ is the same as $$(p-1)!\equiv p-1\pmod p\ $$ In $\pmod p$ negative doesn't mean what it does in your regular everyday arithmetic.

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