Thread: Help with University Algebra Problems

Hello everyone,
Can anyone help me with the problems below, it's very urgent.

1. R is a ring with property that x^2=x for each x in R.

a) Show that each element of R equals its own negative. (Hint: Consider (x+x)^2)

b) Hence show that R is a commutative ring.

2. Find a factorisation of 9+4sqrt(3) into irreducibles, in Z[sqrt(3)]

Thank you for your input.

Originally Posted by Joey123

Hello everyone,
Can anyone help me with the problems below, it's very urgent.

1. R is a ring with property that x^2=x for each x in R.

a) Show that each element of R equals its own negative. (Hint: Consider (x+x)^2)

b) Hence show that R is a commutative ring.

a) We have
But, .
Then, .

b) We have .
On the other side, .
Then .

Thank you so much red_dog for helping me with Q1, I finally understand it now.

Is there anyone who could please help me with Q2 ? This is the last one.

Q2. Find a factorisation of 9+4squareroot(3) into irreducibles, in Z[squareroot(3)]

Your input will be appreciated, Thank you.

Originally Posted by Joey123

Q2. Find a factorisation of 9+4squareroot(3) into irreducibles, in Z[squareroot(3)]

A very bad I see how to procede is to write,

Thus,
.
Thus,
.

And now try to solve for integers. Guessing is the best way it seems. I played around with these for a few minutes but I did not find anything, maybe you do better.