Automorphisms

• Determination of subfields
• Automorphism groups of normal and abelian fields
• Isomorphism of number fields
• Galois group of number fields having degree less than 24
• Galois correspondence
• Ramification theory

Consider the extension K of Q by a root of x⁹ - 57x⁶ + 165x³ - 6859. The maximal order of K is found in 0.25 seconds, the class group ( $\mbox{\bf Z}/3\mbox{\bf Z}$) is found unconditionally in 41 seconds (conditionally, under GRH, in 26 seconds, using even smaller bounds it is possible to compute the class group in 4.11 seconds), the unit group ( $\mbox{\bf Z}/2\mbox{\bf Z}\oplus\mbox{\bf Z}\oplus\mbox{\bf Z}\oplus\mbox{\bf Z}\oplus\mbox{\bf Z}$) in 3.50 seconds and the Galois group of order 18 in 0.25 seconds. The four subfields of degree 3 are found in 0.10 seconds.