# Thread: Help with University Algebra Problems

1. ## 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.

2. 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 $\displaystyle (x+x)^2=x+x$
But, $\displaystyle (x+x)^2=(x+x)(x+x)=x^2+x^2+x^2+x^2=x+x+x+x$.
Then, $\displaystyle x+x+x+x=x+x\Rightarrow x+x=0\Rightarrow x=-x$.

b) We have $\displaystyle (x+y)^2=x+y$.
On the other side, $\displaystyle (x+y)^2=(x+y)(x+y)=x^2+xy+yx+y^2=x+xy+yx+y$.
Then $\displaystyle x+xy+yx+y=x+y\Rightarrow xy+yx=0\Rightarrow xy=-yx=-(-yx)=yx$.

3. 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.

4. 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,
$\displaystyle 9+4\sqrt{3} = (a+b\sqrt{3})(c+d\sqrt{3})$
Thus,
$\displaystyle (ac+3bd)=9$.
Thus,
$\displaystyle (ad+bc)=4$.

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.