Let Q\[\sqrt{2},\sqrt{3}] denote the smallest subring of the complex numbers containing the rational numbers, , and . Is Q[\sqrt{2}+\sqrt{3}] =Q[\sqrt{2},\sqrt{3}]?
How do I go about proving or disproving it? I'm totally lost...
Let Q\[\sqrt{2},\sqrt{3}] denote the smallest subring of the complex numbers containing the rational numbers, , and . Is Q[\sqrt{2}+\sqrt{3}] =Q[\sqrt{2},\sqrt{3}]?
How do I go about proving or disproving it? I'm totally lost...
I think it's true. I mean, unless I'm mistaken, every element of Q[\sqrt{2}, \sqrt{3}] is of the form a + b\sqrt{2} + c\sqrt{3} for a,b,c rational, and every element of Q[\sqrt{2} + \sqrt{3}] is of the form x + y(\sqrt{2} + \sqrt{3}) for x,y rational, but after this point I'm stuck.
there are two ways you might proceed:
one way is to show the sets contain the same elements, equivalently, that each set contains the other.
a slightly more sophisticated way is to show that each ring is generated by the same same set (you should have 3 generators, the choice of which should be fairly obvious).
you are mistaken that every element of Q[√2, √3] is of the form a + b√2 + c √3. for example, √6 is not of this form, yet is obviously in the ring being (√2)(√3).
the key in all cases, is to see that √2+√3 satisfies some polynomial of minimal degree in Q[x]. what will the degree of this polynomial have to be? (hint: it is not of degree 2).
you're trying to prove Q(√2,√3) = Q(√2+√3).
it is obvious that √2+√3 is in Q(√2,√3). what is NOT obvious is that √2 and √3 are in Q(√2+√3).
now (√2+√3)^2 = 5+2√6, that's not terribly helpful.
however, (√2+√3)^3 = (√2)^3 + 3(√12) + 3(√18) + (√3)^3 = 11√2 + 9√3. this IS helpful:
(1/2)((√2+√3)^3 - 9(√2+√3)) = √2 and (1/2)(11(√2+√3) - (√2+√3)^3) = √3.
alternatively, if you know that √2+√3 is a root of x^4 - 10x^2 + 1, then
we know that (√2+√3)(√2+√3)^3 = 10(√2+√3)^2 - 1. working within the complex field, we know we can divide by √2+√3 to get:
(√2+√3)^3 = 10(√2+√3) - (√3 - √2) = 11√2 + 9√3.