Galois Groups

Symmetry of Polynomial Roots

A polynomial equation may possess several roots related by hidden algebraic symmetries. Consider

Its roots are

These roots can be exchanged without affecting any rational number. Indeed, the transformation

preserves all algebraic relations involving rational coefficients. For example,

remains true after the substitution.

Such transformations are called automorphisms of fields. The set of all automorphisms preserving the base field forms the Galois group.

The fundamental idea of Galois theory is that the arithmetic structure of a polynomial is encoded in the symmetries of its roots.

Field Automorphisms

Let $L$ be a field.

Definition. A field automorphism of $L$ is a bijection

satisfying

and

for all $a,b\in L$.

Since multiplication is preserved, inverses are also preserved:

whenever $a\neq 0$.

An automorphism therefore respects the entire field structure.

If $K\subseteq L$, we are usually interested in automorphisms that leave every element of $K$ fixed.

Definition. The Galois group of the extension $L/K$ is

the group of all field automorphisms of $L$ that fix $K$ pointwise.

Thus,

Composition of maps gives the group operation.

First Example

Consider the extension

Every element has the form

An automorphism fixing $\mathbb{Q}$ must send $\sqrt{2}$ to another root of its minimal polynomial

The roots are

Therefore there are exactly two automorphisms:

1. The identity automorphism

2. The conjugation automorphism

Hence

contains two elements.

As a group, it is isomorphic to the cyclic group

The extension degree is also $2$:

This equality between extension degree and group size is a central feature of Galois extensions.

Automorphisms and Polynomial Roots

Let $L$ be the splitting field of a polynomial

Every automorphism in

permutes the roots of $f(x)$.

Indeed, suppose $\alpha$ is a root:

Applying an automorphism $\sigma$, we obtain

Since $\sigma$ fixes the coefficients of $f(x)$,

Thus $\sigma(\alpha)$ is again a root of the polynomial.

Consequently, the Galois group acts as a permutation group on the roots.

This observation connects field theory with group theory.

Example: $x^3-2$

Consider

over $\mathbb{Q}$. Its splitting field is

where $\omega$ is a primitive cube root of unity.

The roots are

Any automorphism fixing $\mathbb{Q}$ must permute these roots.

One possible automorphism sends

while fixing $\omega$. Another sends

Together these generate a group of order $6$, isomorphic to the symmetric group

Thus

The polynomial $x^3-2$ therefore possesses the full symmetry structure of permutations on three objects.

Fixed Fields

Given a subgroup

one may consider the elements fixed by every automorphism in $G$.

Definition. The fixed field of $G$ is

For example, in

the nontrivial automorphism sends

The only elements unchanged are those with $b=0$. Hence the fixed field is precisely

The interaction between subgroups and fixed fields leads to the fundamental theorem of Galois theory.

Galois Extensions

Not every field extension behaves perfectly with respect to automorphisms. The best situation occurs when the extension is both normal and separable.

Definition. A finite extension $L/K$ is called a Galois extension if

1. every irreducible polynomial in $K[x]$ having one root in $L$ splits completely in $L$;
2. every irreducible polynomial has distinct roots.

For fields of characteristic zero, separability is automatic. Thus finite splitting fields over $\mathbb{Q}$ are Galois extensions.

If $L/K$ is Galois, then

This equality reveals that the algebraic size of the extension equals the number of its symmetries.

The Fundamental Theorem of Galois Theory

The central theorem of Galois theory establishes a correspondence between fields and groups.

Let $L/K$ be a finite Galois extension. Then:

• every intermediate field

corresponds to a subgroup

• every subgroup

corresponds to a fixed field

These correspondences reverse inclusion:

Thus algebraic information about field extensions becomes equivalent to group-theoretic information about automorphisms.

This theorem is one of the deepest structural results in mathematics.

Solvability by Radicals

The historical origin of Galois theory was the study of polynomial equations.

Quadratic, cubic, and quartic equations admit formulas involving radicals. However, general quintic equations do not.

The reason lies in group theory.

A polynomial is solvable by radicals precisely when its Galois group is a solvable group.

For example:

• quadratic equations correspond to groups of order $2$;
• general cubic equations correspond to subgroups of $S_3$;
• general quintic equations correspond to $S_5$, which is not solvable.

Thus the impossibility of a general quintic formula is ultimately a theorem about symmetry.

Galois Groups in Number Theory

Galois groups govern much of modern number theory.

Cyclotomic fields, finite fields, elliptic curves, modular forms, and algebraic varieties all possess associated Galois actions. Arithmetic information is frequently encoded in how the absolute Galois group

acts on algebraic objects.

Modern theories such as class field theory and the Langlands program may be viewed as vast generalizations of the correspondence between arithmetic and symmetry first discovered by Évariste Galois.