I reached this long section in my book about Q Fields, and after reading through this section, a couple of the examples have me feeling less than enlightened. Here they are:
1. If alpha = a + b*Sqrt(d) (which exists in) Q[Sqrt(d)], an expression for a/a_bar in terms of a,b, and d can be found, assuming d is not a perfect square. I read this and was like what????
My work thus far: I don't know what they mean by a/a_bar. I know I did Sqrt[(alpha-a)/b)=d and somehow we force d not to be an integer... perhaps they mean a_bar = alpha-a ? No clue
2. Does anyone know if every element of Q[Sqrt(2)] have a square root in Q[Sqrt(2)] ? Can this be proven/disproven with a counterexample?
I just have problems entirely on where to go with that one.
I really appreciate any and all help! Thank you!
2. (Should be fixed now compared with what I initially wrote) This is pretty easily disproven. Consider that
Suppose that . So
Clearly a and b can't both equal 0. Suppose a = 0, then
b is not rational, contradiction.
Suppose b = 0, then a is the square root of 3, also not rational, thus contradiction. QED.