Galois Group of Polynomials

Tricks

Discriminant of x³+px+q

-4p³-27q²

x^4+bx²+d if b²-4d and d are not squares

D_8

x^4+bx²+d if b²-4d is a square and d is not

Z/4

x^4+bx²+d if d is a square

V_4

Discriminant of x^4+cx+d

-27c^4+256d³

Resolvent cubic of x^4+cx+d

x³-4dx-c²

Resolvent Cubic is Irred and D is not a square

S_4

Resolvent Cubic is Irred and D is a square

A_4

Resolvent Cubic splits into 3 linear factors

V_4

If resolvent cubic splits into linear and irred quadratic

x²-r’ and x²-r’x+d

resolvent cubic splits into linear and irred quadratic, both have roots in K(D^1/2)

Z/4

resolvent cubic splits into linear and irred quadratic, one does not have root in K(D^1/2)

D_8