Solving Number Theory: Showing Congruence Has Exactly k Distinct Solutions

[teddyayalew]

Homework Statement

http://i43.tinypic.com/fymy3l.jpg

question 22.4 (a)

Homework Equations

The Attempt at a Solution

x^k=(x^p-1)^m = (x^p-1-1)(1 + x^p-1 +x^2(p-1) +...+x^(m-1)(p-1))

I know that x^p-1-1 = 0 mod p has p-1 solutions but I can't make anything from the geometric sum. Can someone push me in the right direction.

 

[SammyS]

Homework Statement

http://i43.tinypic.com/fymy3l.jpg

question 22.4 (a)

Homework Equations

The Attempt at a Solution

x^k=(x^p-1)^m = (x^p-1-1)(1 + x^p-1 +x^2(p-1) +...+x^(m-1)(p-1))

I know that x^p-1-1 = 0 mod p has p-1 solutions but I can't make anything from the geometric sum. Can someone push me in the right direction.

(x^p-1-1)(1 + x^p-1 +x^2(p-1) +...+x^(m-1)(p-1)) ≠ (x^p-1)^m

(x^p-1-1)(1 + x^p-1 +x^2(p-1) +...+x^(m-1)(p-1)) = (x^p-1)^m - 1