Multiplication in R=\mathbb{Z}_5 \times \mathbb{Z}_5: Operations and Definitions

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Icebreaker
How is multiplication in R=\mathbb{Z}_5 \times \mathbb{Z}_5 defined? if (a,b) and (c,d) is in R, what's (a,b)(c,d)? (ac,bd)?
 
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Usually pointwise (i.e. (a, b)(c, d) = (ac, bd) as you guessed). It can easily be extended to other groups (rings).
 
What do you mean by extending to other rings? I'm trying to find an isomorphism between Z5XZ5 and Z5[x]/X^2+1 and am having a hard time finding it. If I can redefine multiplication in Z5XZ5 then it will be easy.
 
I mean that given any rings G, H, you can easily define the product ring GxH in the same (pointwise) fashion.

I doubt the author (unless he or she said otherwise) intended for you to redefine Z_5 x Z_5.

Given any polynomial p, there are unique constants a, b and a polynomial q such that

p(x) = q(x)(x^2 + 1) + ax + b.

There seems to be an obvious function between Z_5[x]/(x^2 + 1) and Z_5 x Z_5 to try. But I haven't myself.
 

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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