The five quartic labels over ℚ #
One monic integral quartic for each of the five transitive-group labels of degree four, with its
label proved through the quartic decision table of TauCeti.FieldTheory.GaloisGroups.Quartic.Basic.
| polynomial | resolvent cubic | discriminant | label |
|---|---|---|---|
X⁴ + X + 1 | X³ - 4X - 1, irreducible | 229, not a square | S₄ = 4T5 |
X⁴ + 8X + 12 | X³ - 48X - 64, irreducible | 331776 = 576² | A₄ = 4T4 |
X⁴ - 2 | X³ + 8X, one rational root | -2048, not a square | D₄ = 4T3 |
X⁴ + 1 | X³ - 4X, splits completely | 256 = 16² | V₄ = 4T2 |
X⁴ + X³ + X² + X + 1 | X³ - X² - 3X + 2, one root | 125, not a square | C₄ = 4T1 |
The table decides the first, second and fourth rows outright. The third and fifth rows are only
placed in the pair {4T1, 4T3} by the table, and a further datum separates them. For
X⁴ + X³ + X² + X + 1 = Φ₅ the Galois group is (ℤ/5)ˣ of order four, the order of 4T1 and
not of 4T3. For X⁴ - 2 the reduction modulo 7 factors as (X - 2)(X - 5)(X² + 4), so by
Dedekind's theorem the Galois image contains a transposition, which fixes two roots; a group with
the cyclic label 4T1 has as many elements as there are roots, so it acts freely on them and
contains no such element.
Irreducibility over ℚ is proved by Gauss's lemma for X⁴ + X + 1 and X⁴ + 8X + 12, ruling
out integral roots and integral monic quadratic factors by reduction modulo 2 and 5; by the
Eisenstein criterion at 2 for X⁴ - 2; and from the irreducibility of the cyclotomic
polynomials Φ₈ = X⁴ + 1 and Φ₅ over ℚ.
Main results #
TauCeti.hasGaloisLabel_X_pow_four_add_X_add_one:X⁴ + X + 1has label4T5.TauCeti.hasGaloisLabel_X_pow_four_add_eight_mul_X_add_twelve:X⁴ + 8X + 12has label4T4.TauCeti.hasGaloisLabel_X_pow_four_sub_two:X⁴ - 2has label4T3.TauCeti.hasGaloisLabel_X_pow_four_add_one:X⁴ + 1has label4T2.TauCeti.hasGaloisLabel_X_pow_four_add_X_pow_three_add_X_sq_add_X_add_one:X⁴ + X³ + X² + X + 1has label4T1.- The orders
24, 12, 8, 4, 4of the five Galois groups, asTauCeti.natCard_gal_*.
References #
- K. Conrad, Galois groups of cubics and quartics (not in characteristic 2), §3, Examples.
- LMFDB, number fields
4.0.229.1,4.0.5184.1,4.2.2048.1,4.0.256.1and4.0.125.1.
X ^ 4 + X + 1: the symmetric group, 4T5 #
X ^ 4 + X + 1 is irreducible over ℚ: it has no root modulo 2, and a monic quadratic
factor X² + aX + b would force 1 = a³ - 2ab and 1 = a²b - b², which has no solution
modulo 2.
The resolvent cubic of X ^ 4 + X + 1 is X ^ 3 - 4 * X - 1.
The discriminant of X ^ 4 + X + 1 is 229.
The resolvent cubic X ^ 3 - 4 * X - 1 of X ^ 4 + X + 1 is irreducible over ℚ: it has
no root modulo 3, so no integral root.
X ^ 4 + X + 1 has label 4T5: its Galois group over ℚ is the symmetric group on its
four roots. The discriminant 229 is prime, hence not a square, and the resolvent cubic is
irreducible.
The Galois group of X ^ 4 + X + 1 over ℚ has order 24.
X ^ 4 + 8 * X + 12: the alternating group, 4T4 #
X ^ 4 + 8 * X + 12 is irreducible over ℚ: it is positive at every integer, being
(m² - 2)² + 4 (m + 1)² + 4, and a monic quadratic factor X² + aX + b would force
8 = a³ - 2ab and 12 = a²b - b², which has no solution modulo 5.
The resolvent cubic of X ^ 4 + 8 * X + 12 is X ^ 3 - 48 * X - 64.
The discriminant of X ^ 4 + 8 * X + 12 is 331776 = 576 ^ 2.
The resolvent cubic X ^ 3 - 48 * X - 64 of X ^ 4 + 8 * X + 12 is irreducible over ℚ:
it has no root modulo 5, so no integral root.
X ^ 4 + 8 * X + 12 has label 4T4: its Galois group over ℚ is the alternating group
on its four roots. The discriminant 576 ^ 2 is a square and the resolvent cubic is
irreducible.
The Galois group of X ^ 4 + 8 * X + 12 over ℚ has order 12.
X ^ 4 + 1: the Klein four-group, 4T2 #
X ^ 4 + 1 is the eighth cyclotomic polynomial, hence irreducible over ℚ.
The resolvent cubic of X ^ 4 + 1 is X ^ 3 - 4 * X, which splits completely over ℚ.
The discriminant of X ^ 4 + 1 is 256 = 16 ^ 2.
X ^ 4 + 1 has label 4T2: its Galois group over ℚ is the Klein four-group acting
regularly on its four roots, the primitive eighth roots of unity. The discriminant 16 ^ 2 is a
square and the resolvent cubic X ^ 3 - 4 * X has the root 0.
The Galois group of X ^ 4 + 1 over ℚ has order 4.
X ^ 4 + X ^ 3 + X ^ 2 + X + 1: the cyclic group, 4T1 #
X ^ 4 + X ^ 3 + X ^ 2 + X + 1 is the fifth cyclotomic polynomial, hence irreducible over
ℚ.
The Galois group of X ^ 4 + X ^ 3 + X ^ 2 + X + 1 over ℚ has order 4: it is the unit
group of ℤ/5.
The resolvent cubic of X ^ 4 + X ^ 3 + X ^ 2 + X + 1 is X ^ 3 - X ^ 2 - 3 * X + 2, with
the rational root 2.
The discriminant of X ^ 4 + X ^ 3 + X ^ 2 + X + 1 is 125 = 5 ^ 3.
X ^ 4 + X ^ 3 + X ^ 2 + X + 1 has label 4T1: its Galois group over ℚ is cyclic of
order four, acting regularly on the primitive fifth roots of unity. The discriminant 125 is not
a square and the resolvent cubic has the rational root 2, which places the label in
{4T1, 4T3}; the order 4 of the Galois group rules out 4T3, of order 8.
X ^ 4 - 2: the dihedral group, 4T3 #
X ^ 4 - 2 is irreducible over ℚ, by the Eisenstein criterion at the prime 2.
The resolvent cubic of X ^ 4 - 2 is X ^ 3 + 8 * X, with the rational root 0.
The discriminant of X ^ 4 - 2 over ℚ is -2048 = -2 ^ 11.
The discriminant of X ^ 4 - 2 over ℤ is -2048 = -2 ^ 11, so every odd prime is a good
prime for X ^ 4 - 2.
X ^ 4 - 2 does not have cyclic Galois group. Modulo 7 it factors with exactly one
quadratic factor, so by Dedekind's theorem its Galois image contains a transposition, which fixes
two of the four roots. A group with the cyclic label 4T1 has order four and acts freely on the
four roots, so it contains no such element.
X ^ 4 - 2 has label 4T3: its Galois group over ℚ is dihedral of order eight. The
discriminant -2048 is not a square and the resolvent cubic X ^ 3 + 8 * X has the rational root
0, which places the label in {4T1, 4T3}; the transposition exhibited by the factorization
modulo 7 rules out 4T1.
The Galois group of X ^ 4 - 2 over ℚ has order 8.