# Thread: for any prime p and any elements "a" of Zp, show irreducible

1. ## for any prime p and any elements "a" of Zp, show irreducible

The other question im stuck on is :

For any prime p and any element a of Zp, show that x^(P) -a and x^(p) + a are irreducible over Zp

2. Originally Posted by nikie1o2
The other question im stuck on is :

For any prime p and any element a of Zp, show that x^(P) -a and x^(p) + a are irreducible over Zp
they are reducible not irreducible! each is a $p$ power of a polynomial of degree one. can you solve the problem now?

3. hint number 2: what is a^p in Zp?

4. ok so a^p=a I found x^p -a to be reducible but still having trouble with x^p+a . I know x^p+a = (x+a)^p because Zp is a field with Char=p but not sure where to go from there.

5. Originally Posted by nikie1o2
ok so a^p=a I found x^p -a to be reducible but still having trouble with x^p+a . I know x^p+a = (x+a)^p because Zp is a field with Char=p but not sure where to go from there.

You can go from there and have a beer (if you're in USA and if you're over 21...) since you've just proved that x^p + a is a product of p linear

factors and thus reducible, so you're done ( Corona is the best!)

Tonio