There is no canonical multiplication on , so you can use the bijection to lift the multiplication on to . For a different quadratic polynomial, you would get a different ring structure (a field for ).Originally Posted by Treadstone 71
No. Just like multiplication on RxR = C isn't defined by (a,b)(c,d) = (ac, bd) but by (a,c)(b,d) = (ac-bd, ad+bc).Originally Posted by Treadstone 71
Define by , where is the unique representative of degree less than 2 of an element in . This is a bijection. Then define
, where the multiplication on the right is the one in . Then work out what this means in . Compare this to the formula for complex multiplication, and think about an explanation.
Hope this helps.