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.
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.