How do I Show that either f(1)=0 or f(1)=1 ii) Show that either f maps each element of F to 0 or f is injective.

fluther is not the place to answer homework questions. That said, I’m happy to help you. Check your private comments.

by the way, I’m a math PhD student, concentrating in abstract algebra.

Hint: A field has exactly two ideals.

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Remember that a homomorphism basically takes the ring structure with it (see the definition of a homomorphism, and compare to that last sentence). For part (i), maybe you could assume that 1 is not sent to 0 or 1, and see if you can find some kind of contradiction. Try playing with it a little.

Ah… I see what you’re driving at. You are applying field theory to Sudoku! Clever!