What is the Proof for (p-1)!≡±1 (mod p) in Number Theory?

koukou
Messages
5
Reaction score
0
Recall the definition of n! (read n factorial"):
n! = (n)(n-1)(n-2) ….(2)(1) =∏(k)
In both (a) and (b) below, suppose p≥3 is prime.
(a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡±1 (mod p).
(b) Prove that (p-1)!≡±1 (mod p)

Zpx x shoud be above p

a and b looks like some theorem proof
 
Physics news on Phys.org
Hints:
1. factorise x^2-1=(x-1)(x+1).

2. this theorem is called wilson theorem.
 
Hint: (2) can be looked at as a case of a and its inverse. The first part, (1) plays a special role in that.
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
View full post »

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Similar threads

I How to Prove a Congruence Relation for Prime Numbers?
Replies
17
Views
2K
I ##(p^k -1) \equiv X \mbox{(mod p)}## via Wilson's theorem
Replies
2
Views
1K
I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
Replies
48
Views
3K
I Questions about non existence of GCDs vs (coimages, cokernels)
Replies
12
Views
148
I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
1K
I ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
Replies
31
Views
1K
I Trouble understanding an online solution to an exercise in Dummit & Foote
Replies
21
Views
705
I Proof for subgroup -- How prove it is a subgroup of Z^m?
Replies
11
Views
2K
I What Conditions Ensure the Minimal Polynomial of α Divides P in Galois Theory?
Replies
12
Views
2K
I Understanding the Results of gcd(x,n)
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top