How many elements does the following field have? is u=1+i a primative element?

I know that the norm of 2+3i =13, thus there will be 13 elements and this field will be isomorphic to GF(13), But i am not sure how to find the elements.

Printable View

- October 8th 2010, 04:07 PMulysses123Fields and gaussian integers
How many elements does the following field have? is u=1+i a primative element?

I know that the norm of 2+3i =13, thus there will be 13 elements and this field will be isomorphic to GF(13), But i am not sure how to find the elements. - October 8th 2010, 07:44 PMtonio
Do as you do to find the elements of a ring of fractions of a polynomial ring divided by the principal

ideal generated by an irreducible pol.:

1) Divide any element by with remainder (why is this possible?)

2) If , then

...

3) Deduce from the above that every element in the quotient ring can be represented by an element with norm < 13 or zero.

Tonio - October 9th 2010, 07:44 PMulysses123
every element can be represented by one with norm <13, but we dont have to take this representation do we?

Using the element 2+3i and then rotating it by 90 degrees i formed a square with these two vectors then took all the elements in it as my represenatives, then using the element 1+i and generating its powers i found that it only has order 4.However for the second field i havnt been able to deduce if 1+i is a generator because one of the powers gives me an element thats not in the field or any of the representative cosets.So now i'm getting confused.