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Old 2017-02-16, 15:55   #7
Dr Sardonicus
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Feb 2017

497010 Posts

Would there be an easy way to generate polynomials with number fields the same as the cyclotomic polynomial of degree 4 or more generally n-1?

Yes -- given the right software [e.g. Pari-GP], and understanding what it does.

Your polynomial,

f=x^4+7*x^3-x^2-7*x+1 is irreducible with [according to Pari-GP] Galois group D4. I also used Pari to find a "reduced" polynomial defining the same field,

g = x^4 - 2*x^3 - 6*x^2 + 7*x + 11 with D = 5^3 * 41

The polynomial f (mod 3) has a repeated quadratic factor because of the "extraneous" factor 3^4 in the discriminant. The polynomial g (mod 3) splits into 2 distinct quadratic factors.

The splitting field contains the quadratic fields defined by t^2 - t - 1, t^2 - t - 10, and t^2 - t - 51 [the last being a defining polynomial for Q(sqrt(D))]

Both f and g split into quadratic factors in the field defined by t^2 - t - 1; g factors as

(x^2 - x + (-t - 3)) * (x^2 - x + (t - 4)) where t^2 - t - 1 = 0.

This means that f and g (mod p) will factor into two quadratic factors whenever 5 is a quadratic residue (mod p), i.e. p == 1 or -1 (mod 5).
However, whether either quadratic factor splits further (mod p) is a more complicated proposition; it cannot be described by rational integer congruences (this is because D4 is not an Abelian group). For example, one of the quadratic factors (mod 11) splits into linear factors; both quadratic factors (mod 31) remain irreducible; both quadratic factors (mod 131) split into linear factors.
If p is congruent to 2 or 3 (mod 5), g (mod p) either remains irreducible, or splits into irreducible quadratic factors. These cases are not describable with rational integer congruences.

I decline to generate more defining polynomials for the field of 7th roots of unity.

Last fiddled with by Dr Sardonicus on 2017-02-16 at 16:01
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