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## Main Question or Discussion Point

I am stuck on this proof.

Zp[x]/(x^2 + 1) is a field iff p is a prime p = 3 (mod 4)

We're assuming p is odd, so p is either 4m + 1 or 4m + 3.

==>/ let Zp[x]/(x^2 + 1) be a field

I need to find that x^2 + 1 is reducible if p =4m+1

I can see it for Z5, Z13, Z17 for instance but I don't seem to be able to generalize it. Any advice.

<==/ if p = 3 (mod 4), we must show x^2 + 1 is irreducible over Zp

I assume otherwise, then x^2 + 1 = (x+a)(x+b)

gives me, ab ≡ 1 (mod p)

a+b≡ 0(mod p)

where should I go after this?

Thanks

Zp[x]/(x^2 + 1) is a field iff p is a prime p = 3 (mod 4)

We're assuming p is odd, so p is either 4m + 1 or 4m + 3.

==>/ let Zp[x]/(x^2 + 1) be a field

I need to find that x^2 + 1 is reducible if p =4m+1

I can see it for Z5, Z13, Z17 for instance but I don't seem to be able to generalize it. Any advice.

<==/ if p = 3 (mod 4), we must show x^2 + 1 is irreducible over Zp

I assume otherwise, then x^2 + 1 = (x+a)(x+b)

gives me, ab ≡ 1 (mod p)

a+b≡ 0(mod p)

where should I go after this?

Thanks