Hi, I am trying to calculate the number of homomorphisms from one field to another: a) F2 ---> F3 b) Q[X]/(X7 - 3) ---> Q[X]/(X8 + 4X5 - 6X + 2) c) F7 [X] / (X2 + X - 1) ---> F7[X] / (X2 + 1) d) Q( 21/4 ) ---> C Attempt at a solution a) I'm pretty sure there are no homomorphisms between F2 and F3 because if there was a homomorphism f, then f(1+1) = f(0) which does not equal f(1) +f(1) = 2 b) I think I need to see how many roots there are of X7 - 3 in Q[X]/(X8 + 4X5 - 6X + 2) since there is a bijection between that and the number of homomorphisms? c) Similarly here d) For this one I think the answer is four (I'm really not sure) because there is a bijection between K-Homomorphisms and the roots of the minimal polynomial of 21/4 in C, which would be 4. And over fields K-homomorphisms are ring homomorphisms? In all honesty I am pretty stuck, and if anyone could give me any advice that would be fantastic. Thanks in advance.