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Old 2018-07-25, 05:12   #1
carpetpool
 
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"Sam"
Nov 2016

22·79 Posts
Post Families of cyclic cubic fields... and more

It is well known that if K is the splitting field of the polynomial t = x^3 + y*x^2 + (y-3)*x - 1, then K must be a cyclic cubic field, as explained here. This result is known as the "simplest cubic fields". However, I was interested in finding more cyclic cubic families, and I did happen to spot another occurence:

If K is the splitting field of the polynomial T = x^3 - y*x^2 - 9*x + y (assume t is irreducible), then K must be a cyclic cubic field. The proof for this is similar to the one illustrated in the pdf link. The discriminant of t is d = (y^2 - 3*y + 9)^2 while the discriminant of T is D = (y^2 + 27)^2. Both splitting fields of y^2 - 3*y + 9 and y^2 + 27 are isomorphic to the field of Einstein Integers --- a field obtained by adjoining a root of y^2 + y + 1 to the field of real numbers, known as the third cyclotomic field.

Besides these two sole examples, are there any more unique cyclic cubic field "families" or is there perhaps a finite number of instances, and it might be the case that these are the only two instances?

For quartic cyclic field families,

by trial and error, I found that when K is the splitting field of the polynomial T = x^4 + y*x^3 - 6*x^2 - y*x + 1, then K must be a cyclic quartic field. t has the discriminant D = (y^2 + 16)^3.

The same question related to cubic fields is addressed to quartic fields.

Last fiddled with by carpetpool on 2020-04-14 at 21:56
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