# Difficult Abstract Algebra Problem

• October 2nd 2010, 10:26 PM
Oipiah
Difficult Abstract Algebra Problem
Last problem of the problem set and don't even know where to begin:
'let p be an odd prime an n be a positive integer. Use the binomial theorem to show that ((1+p)^p)^(n-1)==1(mod p^n) but ((1+p)^p)^(n-2) is not equivalent to 1(mod p^n). Deduce that 1+p is an element of order p^(n-1) in the multiplicative group z/pz.'
I wrote the expansion of both of these using the Binomial Theorem, but can't seem to parse anything from that. The last part is easy since it directly follows from the first part, but I don't even know where to start the first part. Any help/ ideas greatly appreciated
• October 4th 2010, 05:14 AM
tonio
Quote:

Originally Posted by Oipiah
Last problem of the problem set and don't even know where to begin:
'let p be an odd prime an n be a positive integer. Use the binomial theorem to show that ((1+p)^p)^(n-1)==1(mod p^n)

This is false, as can be seen from the example $64^3=\left((1+3)^3\right)^{4-1}\neq 1\!\!\pmod {3^4}$.

Perhaps you meant $(1+p)^{p^{n-1}}=1\!\!\pmod{p^n}$ , though you wrote two parenthese at each side on the LHS ?

Tonio

but ((1+p)^p)^(n-2) is not equivalent to 1(mod p^n). Deduce that 1+p is an element of order p^(n-1) in the multiplicative group z/pz.'
I wrote the expansion of both of these using the Binomial Theorem, but can't seem to parse anything from that. The last part is easy since it directly follows from the first part, but I don't even know where to start the first part. Any help/ ideas greatly appreciated

.