Q[ ] Problems, Lots of good examples! (Need some help!)
Okay, so I finally reached this long section in my book about all of these Q[ ]'s. After reading through this section, a couple of the examples have me miffed.
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????
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?
3. There exists a defining equation for the golden ratio: (1+Sqrt(5))/2 , and its norm in Q[Sqrt(5)] can also be found. Can anyone help me find this equation and the norm?
4. Assuming a and b are elements of Q[Sqrt(d)],, it can be shown that N(ab)=N(a)N(b) and N(a/b)=N(a)/N(b). How can this be shown? Should a exist in Q, how can we show that N(a)=(a^2) ?
5. There exists integers 'd' does the field Q[Sqrt(d)] have elements a with negative norm N(a), assuming that d is not a perfect square. Does anyone know what this 'd' is and how to prove this?
6. Examining Q[Sqrt(-1)], an equation can be written relating N(a) to |a| where | | means the natural absolute value defined for complex numbers. What is this equation and for which Q[Sqrt(d)] would this formula be correcT?
I really appreciate all the help! (Sorry I don't know how to use latex by the way!)