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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 27 23:26 EDT 2020. Contains 338047 sequences. (Running on oeis4.)