# Understand the problem

Find the number of ring homomorphism from to

##### Source of the problem

TIFR 2019 GS Part A, Problem 16

##### Topic

Algebra

##### Difficulty Level

Moderate

##### Suggested Book

Abstract Algebra, Dummit and Foote

# Start with hints

Do you really need a hint? Try it first!

Now observe that in . Then can you think about the ring ?

Then we have to find the number of ring homomorphism from to . How many elements does the ring have?

. It is not a field but it has elements as .

#### Now what is the next move?

For any commutative ring there is a bijection between the homomorphisms and the elements in . To any , and determines the homomorphism.

Hence, if one counts the number of elements in , the number of homomorphisms you want is just the square.

Now, has elements.

#### So, the number of ring homomorphisms is .

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