Galois Theory

Galois Theory represents one of the crowning achievements of algebra, providing a bridge between field theory and group theory. By studying the symmetry of the roots of a polynomial, we can determine whether that polynomial can be solved using radicals—a problem that remained open for centuries.

Field Automorphisms

Let $L$ be an extension of $K$. An automorphism of $L$ is an isomorphism $\sigma :L\to L$. We are interested in the set of automorphisms that fix every element of the base field $K$: $\text{Aut}(L/K)=\{\sigma \in \text{Aut}(L)\mid \sigma (k)=k\text{ for all }k\in K\}$ This set forms a group under composition.

The Galois Group

If the extension $L/K$ is both normal (it is a splitting field) and separable (all roots of irreducible factors are distinct), it is called a Galois extension. In this case, $\text{Aut}(L/K)$ is called the Galois Group, denoted $\text{Gal}(L/K)$.

Normal and Separable Extensions

To apply Galois Theory, an extension must satisfy two technical conditions:

1. Normal: Every irreducible polynomial in $K[x]$ that has a root in $L$ must split completely in $L$. This ensures that $L$ contains “all the symmetry” of the roots.
2. Separable: Every irreducible polynomial in $K[x]$ has distinct roots in an algebraic closure. Over fields of characteristic 0 (like $\mathbb{Q}$), all irreducible polynomials are separable.

The Fundamental Theorem of Galois Theory

The power of Galois Theory lies in the Galois Correspondence. Given a finite Galois extension $L/K$ with Galois group $G=\text{Gal}(L/K)$, there is a one-to-one inclusion-reversing correspondence between:

• Subgroups $H$ of $G$.
• Intermediate fields $E$ such that $K\subseteq E\subseteq L$.

Key Properties:

• The subfield corresponding to a subgroup $H$ is the fixed field: $L^H=\{\alpha \in L\mid \sigma (\alpha )=\alpha \text{ for all }\sigma \in H\}$.
• The degree of the extension is equal to the size of the group: $[L:K]=|G|$.
• An intermediate extension $E/K$ is normal if and only if its corresponding subgroup $H$ is a normal subgroup of $G$.

Solvability by Radicals

For centuries, mathematicians sought formulas for the roots of quintic (5th degree) equations, similar to the quadratic formula. Galois Theory proved that this is impossible.

• A polynomial is solvable by radicals if its roots can be expressed using field operations and $n$-th roots.
• Abel-Ruffini Theorem: A polynomial is solvable by radicals if and only if its Galois group is a solvable group.
• Since the symmetric group $S_5$ (and higher) is not solvable, the general quintic equation cannot be solved by radicals.

Cyclotomic Extensions

Cyclotomic fields are generated by the $n$-th roots of unity, ${\zeta }_n=e^{2\pi i/n}$.

• The extension $\mathbb{Q}({\zeta }_n)/\mathbb{Q}$ is Galois.
• The Galois group $\text{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})$ is isomorphic to $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, the multiplicative group of integers modulo $n$.
• These fields are fundamental in number theory and the study of Fermat’s Last Theorem.

Python: Computing Permutations of Roots

We can use Python to visualize how automorphisms permute the roots of a polynomial.

import itertools

def is_automorphism(perm, relations):
    """
    Checks if a permutation of roots preserves the polynomial relations.
    (Simplified conceptual model)
    """
    for rel in relations:
        permuted_rel = [perm[i] for i in rel]
        if permuted_rel not in relations:
            return False
    return True

# Roots of x^4 - 2: [a, -a, ai, -ai]
roots = ['a', '-a', 'ai', '-ai']
# Possible symmetries (e.g., complex conjugation)
# This is a complex topic but we can identify the group D8
perms = list(itertools.permutations(range(4)))
print(f"Total permutations: {len(perms)}")
# Galois theory filters these to find the true symmetries.

Significance

Galois Theory turned the problem of finding roots into a problem of studying group structures. It provided the ultimate proof that math is about structure rather than just calculation. It also laid the groundwork for modern algebraic geometry and representation theory.

Exercise