Math Help - Mixed Bag

1. 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.

2. Originally Posted by jdizzle1985
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.

3. Originally Posted by jdizzle1985
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.