 TOPICS  # Galois Group

Let be an extension field of , denoted , and let be the set of automorphisms of , that is, the set of automorphisms of such that for every , so that is fixed. Then is a group of transformations of , called the Galois group of . 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 number field .

The Galois group of consists of the identity element and complex conjugation. These functions both take a given real to the same real.

Abhyankar's Conjecture, Finite Group, Fundamental Theorem of Galois Theory, Galois's Theorem, Galois Theory, Group, Solvable Group, Symmetric Group

Portions of this entry contributed by David Terr

## Explore with Wolfram|Alpha More things to try:

## References

Birkhoff, G. and Mac Lane, S. "The Galois Group." §15.2 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 397-401, 1996.Jacobson, N. Basic Algebra I, 2nd ed. New York: W. H. Freeman, p. 234, 1985.

Galois Group

## Cite this as:

Terr, David and Weisstein, Eric W. "Galois Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaloisGroup.html