Galois Theory HT25, Quartic equations

• [[Course - Galois Theory HT25]]^U
  • [[Notes - Galois Theory HT25, Computing the Galois group]]^U
  • [[Notes - Galois Theory HT25, Cubic equations]]^U
  • [[Notes - Galois Theory HT25, Quintic equations]]^U

An (almost) complete characterisation of the Galois group in terms of the roots

This is not covered in too much detail in the course notes, but there is almost a complete characterisation of the Galois group of any quartic just in terms of expressions in the roots. This comes from [[Algebra, Artin]]^N.

Suppose $f(x)$ has roots $\alpha _ 1, \alpha _ 2, \alpha _ 3, \alpha _ 4$ and let $\Delta$ be the discriminant of $f(x)$.

Let

• $\beta _ 1 = \alpha _ 1 \alpha _ 2 + \alpha _ 3 \alpha _ 4$
• $\beta _ 2 = \alpha _ 1 \alpha _ 3 + \alpha _ 2 \alpha _ 4$
• $\beta _ 3 = \alpha _ 1 \alpha _ 4 + \alpha _ 2 \alpha _ 3$

and define the resolvent cubic as $g(x) = (x - \beta _ 1)(x - \beta _ 2)(x - \beta _ 3)$. Then we have the characterisation: