Simfish
Feb20-08, 09:34 AM
Determine the irreducible polynomial for $\alpha$ = $\sqrt {3} + \sqrt {5}$ over each of the following fields.
(c) Q($\sqrt {10}$) (d) Q($\sqrt {15}$)
So I'm having trouble with both c and d. I can try $(x - \sqrt {3} - \sqrt {5})^2$ or $x^2 - (\sqrt {3} + \sqrt {5})^2$ but it's quite difficult to find irreducible polynomials for those fields this way (I can do it for $\sqrt {3}$ easily.
(c) Q($\sqrt {10}$) (d) Q($\sqrt {15}$)
So I'm having trouble with both c and d. I can try $(x - \sqrt {3} - \sqrt {5})^2$ or $x^2 - (\sqrt {3} + \sqrt {5})^2$ but it's quite difficult to find irreducible polynomials for those fields this way (I can do it for $\sqrt {3}$ easily.