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I am trying to calculate the number of homomorphisms from one field to another:

a) F_{2}---> F_{3}

b) Q[X]/(X^{7}- 3) ---> Q[X]/(X^{8}+ 4X^{5}- 6X + 2)

c) F_{7}[X] / (X^{2}+ X - 1) ---> F_{7}[X] / (X^{2}+ 1)

d) Q( 2^{1/4}) ---> C

Attempt at a solution

a) I'm pretty sure there are no homomorphisms between F_{2}and F_{3}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 X^{7}- 3 in Q[X]/(X^{8}+ 4X^{5}- 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 2^{1/4}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.

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# Number of Homomorphisms

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