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Math Help - Mixed Bag

  1. #1
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    Mixed Bag

    Hi guys, i've been struggling with a few bits and pieces from a homework sheet and have no idea how to present the following q's. I'm a final year undergrad. Can anyone provide me with solutions to the following?

    1.Show that if gcd(a, c) = gcd(b, c) = 1 then gcd(ab, c) = 1

    2.The equation X^2 + Y ^2 = 3Z^2 has no non trivial solution
    (i.e. 6= (0, 0, 0)). Show the same for X^2 + Y ^2 = 11Z^2.

    3.Find a formula giving all the solutions to the equation X^2 + 2Y^2 = Z^2 with X,
    Y and Z in N and gcd(X, Y, Z) = 1.

    4.Determine whether or not the following polynomials are irreducible in Q[X] X^3 +2X^2 +4X +2.
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  2. #2
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    Quote Originally Posted by jdizzle1985 View Post
    1.Show that if gcd(a, c) = gcd(b, c) = 1 then gcd(ab, c) = 1
    Let a,b,c>0. Let d=\gcd(ab,c). Then d|ab and d|c. Now \gcd(a,d) = 1 because \gcd (a,c)=1. Thus d|b but then d=1 because \gcd(b,c)=1.

    4.Determine whether or not the following polynomials are irreducible in Q[X] X^3 +2X^2 +4X +2.
    It is a degree 3 polynomial so you can test if it has any rational roots.
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  3. #3
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    Quote Originally Posted by jdizzle1985 View Post
    2.The equation X^2 + Y ^2 = 3Z^2 has no non trivial solution
    Both x,y cannot be even. And both x,y cannot be odd for then a square of an odd integer has form 4k+1 so in sum the LHS would have form 4k+2 while RHS would have form 4k. Thus x is even and y is odd WLOG. But then LHS has form 4k+1 but RHS would have form 4k+3. Thus there are no solutions.
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