Hallo,
Never say this before you write the exam(which I'm going to fail)
1 is not a root, but -1 is. I don't know if it helps...Would this just be the identity element 1?
I don't get the thing with the subspace F... sorry
Sorry for all the algebra questions everybody, I have my exam on Wednesday (which I'm going to fail) and seem to be struggling with everything as it's so abstract...
Let
Let F be the smallest subfield of R containing a root of g. What are the elements of F?
Would this just be the identity element 1?
What is the dimension of F considered as a vector space over the rationals?
This I just do not understand at all....can anyone please explain?
Thanks in advance!
Ok cheers for the relpy Moo, think I'll just pray that those questions don't come up!
I've got another question which seems to come up every year, seems a little easier but I still don't see what to do?!
Write down the inverse of in the form where are all rational numbers.
Can anyone please explain the method for this?
Thanks again.
Note that any field should contain 0 and 1. Here it should also contain according to the posed question. Now wonder what happens . Since closure would force them into the field. The real thing you should observe here is that :
-----------------------------------(*)
This means every power of in this field can be brought down to the form
For example:
Now use the property of we just proved in (*)
So solve this system of equations:
Unless I have been careless in my arithmetic you should get the right answer
Clearly it is 3. Since is the basis.
OK I've just worked through this example, and understood everything...EXCEPT where the comes from? Obviously I see that it comes from rearranging, but what I mean is do you always have to know what g is before you can find this alpha equation?
Also there were a few signs wrong in your working, but got the answers to be
Is there a way to check these answers correct after doing this method?
Thanks again
Yes! The field generated is due to a particular polynomial. Its called the generating polynomial of the field.
Sorry for that... I did not work it out on paper
Yes multiply out the obtained answer with and check if you get 1