Prove that this Function is a Homomorphism

[peelgie]

Summary:: Abstract algebra

I have a problem with this task. Please help.

[Moderator's note: Moved from a technical forum and thus no template.]

 

[fresh_42]

Can you explain what the terms mean, and what $\mathbb{Z}_{31}^*$ is?
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[peelgie]

$$(Z_{31}^{*} = \{1, 2, 3, ...,30\},\cdot_{31}) $$

 

[fresh_42]

You have to show some efforts so that we can see where your problems are. We will not do the homework for you.

 

[peelgie]

(1)
I need to prove this equation:
$$
\varphi(x\cdot_{31}y) = \varphi(x)\cdot_{31}\varphi(y)
$$
So:
$$
\varphi(x\cdot_{31}y) = (x\cdot_{31}y)^{18} = x^{18}\cdot_{31}y^{18} = \varphi(x)\cdot_{31}\varphi(y)
$$ That is Correct? Function is homomorphism?

 

[fresh_42]

(1)
I need to prove this equation:
$$
\varphi(x\cdot_{31}y) = \varphi(x)\cdot_{31}\varphi(y)
$$
So:
$$
\varphi(x\cdot_{31}y) = (x\cdot_{31}y)^{18} = x^{18}\cdot_{31}y^{18} = \varphi(x)\cdot_{31}\varphi(y)
$$ That is Correct? Function is homomorphism?

This is correct, but it could be that you have to justify the equation in the middle: $(x\cdot_{31}y)^{18} = x^{18}\cdot_{31}y^{18}$. It depends on what you may use and what not. Since you haven't told us this information, we cannot know.

I mean "trivial" is also a valid answer. It all depends on what can be assumed as given and what cannot.

 

[LCKurtz]

What is the $^*31$ operation?

 

[peelgie]

mod 31

 

[fresh_42]

mod 31

Yes, sure. But why is it a ring homomorphism?