2
$\begingroup$

For $x^2+x+1$, there exist infinitely many integers $w$ such that $x^2+x+1+w$ defines the same field as $x^2+x+1$.

They are $w$ $=$ $6, 60, 546, 4920,...$ There is a closed form for odd $k$, namely $w$ $=$ $((3^k+1)/4)$$-1$

I can't seem to prove / disprove this for the fact of weather there are infinitely many integers $w$ such that $x^4+x^3+x^2+x+1+w$ defines the same field as $x^4+x^3+x^2+x+1$.

The same with weather there are infinitely many integers defining $w$ in $x^6+x^5+x^4+x^3+x^2+x+1+w$ defines the same field as $x^6+x^5+x^4+x^3+x^2+x+1$.

If there is at least one integer $w$ for degrees $4$ or $6$, please post them as a counterexample.

  • 0
    Are you looking at this modulo some prime or what? I have this suspicion that the [tag:finite-fields] tag is mistakenly included. Because you discuss integers it seems to me that your fields are infinite (because they contain $\Bbb{Q}$, an infinite set, as a subfield). A cool question nevertheless. – 2017-02-26
  • 0
    @JyrkiLahtonen for the nth cyclotomic polynomial defined in field Zn – 2017-02-27

0 Answers 0