This is the smallest instance of a non-solvable Galois group, implying that the general quintic cannot be solvable in radicals and motivating the creation of Galois theory.
Group action invariants
Degree $n$: | $5$ | |
Transitive number $t$: | $4$ | |
Group: | $A_5$ | |
CHM label: | $A5$ | |
Parity: | $1$ | |
Primitive: | yes | |
Nilpotency class: | $-1$ (not nilpotent) | |
$\card{\Aut(F/K)}$: | $1$ | |
Generators: | (1,2,3), (3,4,5) |
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
Low degree siblings
6T12, 10T7, 12T33, 15T5, 20T15, 30T9Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 3, 1, 1 $ | $20$ | $3$ | $(3,4,5)$ |
$ 2, 2, 1 $ | $15$ | $2$ | $(2,3)(4,5)$ |
$ 5 $ | $12$ | $5$ | $(1,2,3,4,5)$ |
$ 5 $ | $12$ | $5$ | $(1,2,3,5,4)$ |
Group invariants
Order: | $60=2^{2} \cdot 3 \cdot 5$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | no | |
Label: | 60.5 |
Character table: |
2 2 . 2 . . 3 1 1 . . . 5 1 . . 1 1 1a 3a 2a 5a 5b 2P 1a 3a 1a 5b 5a 3P 1a 1a 2a 5b 5a 5P 1a 3a 2a 1a 1a X.1 1 1 1 1 1 X.2 3 . -1 A *A X.3 3 . -1 *A A X.4 4 1 . -1 -1 X.5 5 -1 1 . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |