Multiplicative functions and homomorphisms

ehrenfest
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Homework Statement


What is the difference between multiplicative functions and homomorphisms?


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The Attempt at a Solution

 
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Multiplicative functions: the product of the image is the image of the product. There is another definition requiring that the two elements must be relatively prime.

Homomorphisms: preserve structure between the domain and the codomain, this often requires a homomorphism to be multiplicative.
 
I don't like that expression "preserve structure between the domain and the codomain" because it seems like it is not mathematical or even objective...

What does that mean in mathematical terms?
 
In rings it means that multiplication and addition in each ring are very similar.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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