P-adic integers and roots of unity

• Nov 21st 2013, 05:43 PM
lahuxixi
P-adic integers and roots of unity
Show that Zp contains all the (p-1)th roots of unity.
For which primes p does Zp contains primitive fourth roots of unity.
Here Zp is the set of p-adic integers.
Proving that it has a (p-1)th root of unity is easy, but ALL roots is another matter. Please help me with these questions..
I think for the second question, p has to be 5, but maybe there are other answers that i didn't think of.
• Nov 21st 2013, 07:20 PM
lahuxixi
Re: P-adic integers and roots of unity
This is what i have done
Let f(x)=x^(n-1)-1
there exist x such that gcd(x,p)=1, just let x=1 and x^(n-1)=1 mod p.
f'(1)=/=0 since it's still 1. so there exist a solution in Zp that f(x)=0
And part 2, since with part 1, Zp has all (p-1)th root, it's obvious that if p=5, Zp had all 4th roots, one of them is a primitive root. But that answer is too stupid
• Nov 21st 2013, 07:56 PM
lahuxixi
Re: P-adic integers and roots of unity
solved