finite field of odd characteristic

[Guest rainpurple]

Question: Let F be a finite field of odd characteristic. Prove that the rings F[X]/(X^2 - a) as a ranges over all nonzero elements of F fall into exactly two isomorphism classes.

 

I am thinking to classify into two cases: a is a perfect square in F and a is not a perfect square. But I don't know what the condition "the character is odd (shou be odd prime?!)" will do here.

 

Thx a lot!

 

Rp~

[matt grime]

The characteristic is always a prime (if it's non-zero), you don't need to state that, so they don't. You need odd characteristic because squaring in charactersitc two does odd things.: (X+r)^"=X^2+r^2