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Math Help - Help with University Algebra Problems

  1. #1
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    Question 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.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Quote Originally Posted by Joey123 View Post
    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 (x+x)^2=x+x
    But, (x+x)^2=(x+x)(x+x)=x^2+x^2+x^2+x^2=x+x+x+x.
    Then, x+x+x+x=x+x\Rightarrow x+x=0\Rightarrow x=-x.

    b) We have (x+y)^2=x+y.
    On the other side, (x+y)^2=(x+y)(x+y)=x^2+xy+yx+y^2=x+xy+yx+y.
    Then x+xy+yx+y=x+y\Rightarrow xy+yx=0\Rightarrow xy=-yx=-(-yx)=yx.
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  3. #3
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    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.
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  4. #4
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    Quote Originally Posted by Joey123 View Post
    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,
    9+4\sqrt{3} = (a+b\sqrt{3})(c+d\sqrt{3})
    Thus,
    (ac+3bd)=9.
    Thus,
    (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.
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