A Galois group of an extension field 
is the group 
of automorphisms 
of 
such that 
for every 
. Thus 
is fixed. The Galois group of 
is denoted 
or 
.
Let 
be a rational
polynomial of degree 
and let 
be the splitting field of 
over 
, i.e., the smallest subfield
of 
containing all the roots of 
. Then each element of the Galois group 
permutes the roots of 
in a unique way. Thus 
can be identified with a subgroup
of the symmetric group 
, the group of permutations of the roots of 
. If 
is irreducible, then 
is a transitive subgroup
of 
, i.e., given two roots 
and 
of 
, there exists an element 
of 
such that 
.
The roots of 
are solvable by radicals iff 
is a solvable group. Since
all subgroups of 
with 
are solvable, the roots of all polynomials of degree
up to 4 are solvable by radicals. However, polynomials of degree 5 or greater are
generally not solvable by radicals since 
(and the alternating group 
) are not solvable for 
.
The inverse Galois problem asks whether every finite group 
is isomorphic to a Galois
group 
for some finite Galois extension field 
. The problem remains open in general.
Huang et al. (2026) proved that the Mathieu
group M23 occurs in this way, completing the realization of
all 26 sporadic groups over 
.
The Galois group of 
consists of the identity element and complex
conjugation. These functions both take a given real
to the same real.
Is a Galois Group over 
." Aug. 8, 2026.