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A075793

Primes p such that f_p(x)=(1296+432*x+108*x^2+24*x^3+5*x^4+x^5) mod p factors as product of 3 linear and one irreducible quadratic factor.

101, 127, 137, 307, 379, 487, 571, 617, 643, 701, 761, 859, 881, 1013, 1039, 1217, 1229, 1231, 1277, 1361, 1447, 1831, 2081, 2179, 2239, 2417, 2467, 2477, 2621, 2861, 2971, 3257, 3413, 3449, 3461, 3559, 3583, 3701, 3907, 4013, 4049, 4133, 4219, 4241

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OFFSET

1,1

COMMENTS

The fact that f_101 factors as a product of 3 linear and one irreducible quadratic factor shows that the Galois group of f(x) is (isomorphic to) the Symmetric group on 5 letters, S_5. That is also the Galois group of 1+2*x+3*x^2+4*x^3+5*x^4+6*x^5.

REFERENCES

N. Jacobson, Basic Algebra I, Freeman and Co, (1985), pp. 301-304.

LINKS

Table of n, a(n) for n=1..44.

EXAMPLE

When p=101, f_p(x)=(x+40)*(x+30)*(x+49)*(x^2+88*x+72) mod p

CROSSREFS

Sequence in context: A135602 A095635 A060916 * A319049 A052086 A154270

Adjacent sequences: A075790 A075791 A075792 * A075794 A075795 A075796

KEYWORD

easy,nonn

AUTHOR

Waldeck Schutzer, Oct 13 2002

STATUS

approved

URL: https://oeis.org/A075793

⇱ A075793 - OEIS