Thread: Q[ ] Problems, Lots of good examples! (Need some help!)

Hello all,

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!)

Originally Posted by Samson

Hello all,

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!)

Too many question, zero self work done. This isn't serious. Show at least some effort.

Tonio

Originally Posted by tonio

Too many question, zero self work done. This isn't serious. Show at least some effort.

Tonio

If you look how long its been since I last posted some questions, its been nearly 4 weeks! During that time I've covered 3 chapters and was able to handle every example and problem on my own. I happened to find this chapter exceptionally difficult, and I've tried it multiple multiple times on my own but I can't make these relations. I came to get some help!

Can anybody help with these? I really appreciate it!

Originally Posted by Samson

Can anybody help with these? I really appreciate it!

Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. It is better for forum organization and better for you to get your questions answered in a more timely manner if you start new threads as necessary for remaining questions. eg. If you have five questions, post two of them in two threads and start a new thread for the remaining one etc.

(You were told this a few posts ago, as well as asked to show some attempt. That is most likely why there has not been any response. I'm certain Tonio was ready to give some help, all you had to do was meet him/her part of the way.)

Thread closed. Please re-post, following the above advice.