Galois Theory Fundamentals

Galois Theory is one of the most elegant and powerful achievements in abstract algebra, creating a profound bridge between two seemingly distinct worlds: field theory and group theory. Its initial motivation—to determine which polynomial equations can be solved by radicals—led to a complete solution of a centuries-old problem and introduced a framework that now underpins modern number theory and algebraic geometry. By translating questions about fields into questions about groups, it provides a precise language for understanding symmetry in algebraic systems.

Fields, Extensions, and Automorphisms

To understand Galois theory, you must first be comfortable with the idea of a field extension. A field extension, denoted $L/K$, is a pair of fields where $K$ is a subfield of $L$. You can think of $L$ as a larger field built by "adjoining" new elements (like $\sqrt{2}$ or $i$) to the smaller base field $K$. The degree of the extension, $[L:K]$, measures its size as the dimension of $L$ when viewed as a vector space over $K$.

The central actors in our story are automorphisms. An automorphism of a field $L$ is a bijective map from $L$ to itself that preserves the field operations (addition and multiplication). For a given extension $L/K$, we are particularly interested in those automorphisms of $L$ that fix $K$ pointwise. This means that for every element $a$ in $K$ and every such automorphism $\sigma$, we have $\sigma (a)=a$. These automorphisms are the symmetries of the extension $L/K$; they shuffle the newly adjoined elements while leaving the familiar ground field untouched. For example, in the extension $\mathbb{C}/\mathbb{R}$, the complex conjugation map, sending $a+bi$ to $a-bi$, is an automorphism that fixes every real number.

Defining Galois Extensions

Not every field extension has a rich enough structure to support a deep Galois correspondence. The special extensions that do are called Galois extensions. This definition is the careful intersection of two crucial properties: normality and separability.

An algebraic extension $L/K$ is normal if every irreducible polynomial in $K[x]$ that has at least one root in $L$ splits completely into linear factors over $L$. Intuitively, if one new element (a root of a polynomial) joins $L$, then all of that polynomial's roots are forced to come along. This ensures the field is "complete" with respect to the polynomials that define it.

An algebraic extension $L/K$ is separable if the minimal polynomial over $K$ of every element in $L$ has distinct roots (i.e., no repeated roots in an algebraic closure). In characteristic zero (like extensions of $\mathbb{Q}$ or $\mathbb{R}$), every algebraic extension is automatically separable. This condition guarantees that elements are not "glued together" in a way that limits automorphisms.

A Galois extension is a field extension that is both normal and separable. This combination guarantees that the number of $K$-fixing automorphisms of $L$ is exactly equal to the degree of the extension: $|\text{Aut}(L/K)|=[L:K]$. This equality is the linchpin that makes the entire theory work.

The Galois Group

For a Galois extension $L/K$, we define its Galois group, denoted $\text{Gal}(L/K)$. This is the set of all field automorphisms of $L$ that fix $K$, equipped with the operation of function composition. Because $L/K$ is Galois, the order of this group (its number of elements) equals the degree $[L:K]$.

The structure of the Galois group encodes the algebraic relations among the adjoined elements. Consider the classic example $L=\mathbb{Q}(\sqrt{2})/K=\mathbb{Q}$. This is a Galois extension of degree 2. The Galois group $\text{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q})$ consists of two automorphisms: the identity map, and the map that sends $\sqrt{2}$ to $-\sqrt{2}$. This group is isomorphic to the cyclic group of order 2, $C_2$. The non-identity automorphism "swaps" the two roots of the polynomial $x^2-2$, which defines the extension.

The Fundamental Theorem of Galois Theory

This theorem is the cornerstone of the subject. It establishes a precise, inclusion-reversing correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group. Let $L/K$ be a finite Galois extension with Galois group $G=\text{Gal}(L/K)$.

The Fundamental Theorem states:

1. Correspondence: There is a one-to-one correspondence between intermediate fields $E$ (with $K\subseteq E\subseteq L$) and subgroups $H$ of $G$.
2. Mapping Down (Field to Subgroup): Given an intermediate field $E$, the corresponding subgroup is $H=\text{Gal}(L/E)$, the set of automorphisms in $G$ that fix the larger field $E$.
3. Mapping Up (Subgroup to Field): Given a subgroup $H$ of $G$, the corresponding intermediate field is the fixed field $L^H=\{x\in L\mid \sigma (x)=x\text{ for all }\sigma \in H\}$.
4. Inclusion-Reversal: If $E_1$ and $E_2$ correspond to subgroups $H_1$ and $H_2$, then $E_1\subseteq E_2$ if and only if $H_2\subseteq H_1$.
5. Degree/Index Relationship: $[E:K]=[G:H]$ and $[L:E]=|H|$, where $[G:H]$ is the index of $H$ in $G$.
6. Normality of Subfields: An intermediate field $E$ is itself a Galois extension over $K$ if and only if its corresponding subgroup $H$ is a normal subgroup of $G$. In this case, $\text{Gal}(E/K)\cong G/H$.

This theorem allows you to analyze the lattice of intermediate fields—often an infinite, complicated task—by studying the finite, structured lattice of subgroups of a group.

Application: Solvability by Radicals

The original application of Galois theory was to the problem of solving polynomial equations with radicals (i.e., using roots of any order, combined with basic arithmetic). The theory translates this problem into a problem about groups.

A polynomial $f(x)\in K[x]$ is solvable by radicals if all its roots can be expressed using elements of $K$ combined via the operations of addition, subtraction, multiplication, division, and taking $n$-th roots. The associated object is the splitting field $L$ of $f$, the smallest Galois extension of $K$ containing all roots of $f$. Its Galois group, $G=\text{Gal}(L/K)$, is called the Galois group of the polynomial.

The breakthrough came by defining a solvable group. A group is solvable if it has a sequence of subgroups, each normal in the previous, with abelian quotients. Galois proved the following definitive theorem:

A polynomial over a field of characteristic zero is solvable by radicals if and only if its Galois group is a solvable group.

This explains the classical results: The quadratic, cubic, and quartic formulas exist because the symmetric groups $S_2$, $S_3$, and $S_4$ are solvable. The Abel-Ruffini theorem, which states that a general polynomial of degree five or higher is not solvable by radicals, follows from the fact that the symmetric group $S_n$ for $n\ge 5$ is not a solvable group. The simple, non-abelian structure of $A_5$ (the alternating group on five letters) blocks the existence of a universal radical formula.

Common Pitfalls

1. Assuming Every Extension is Galois: A common error is to call $\text{Aut}(L/K)$ the Galois group for a non-Galois extension. Remember, the term "Galois group" is reserved for when $L/K$ is a Galois (normal and separable) extension. For a non-Galois extension, the group of automorphisms may be smaller than the degree. Always verify normality and separability first.
2. Misidentifying the Fixed Field: When applying the Fundamental Theorem, ensure you correctly compute the fixed field $L^H$. It is not just the elements fixed by a generator of $H$, but the elements fixed by every automorphism in the subgroup $H$. You must check the condition against all elements of $H$.
3. Confusing the Correspondence Direction: The correspondence is inclusion-reversing. A larger intermediate field corresponds to a smaller subgroup. A useful mnemonic: more elements in the field means fewer automorphisms are allowed (they have more to fix), so the corresponding subgroup is smaller.
4. Overlooking Separability in Positive Characteristic: When working with fields of characteristic $p$ (like finite fields), separability is not automatic. An extension like ${\mathbb{F}}_p(t^{1/p})/{\mathbb{F}}_p(t)$ is purely inseparable and not Galois, despite potentially being normal. Failing to check separability in such contexts is a frequent oversight.

Summary

• Galois Theory establishes a fundamental bridge between field theory and group theory by studying the symmetries (automorphisms) of field extensions.
• A Galois extension $L/K$ is a normal and separable field extension; its Galois group $\text{Gal}(L/K)$ is the group of all automorphisms of $L$ that fix $K$, and its order equals the degree $[L:K]$.
• The Fundamental Theorem of Galois Theory provides a precise, inclusion-reversing bijection between the intermediate fields of a Galois extension and the subgroups of its Galois group, linking field degrees to group indices.
• The historical problem of solvability by radicals is completely resolved by this theory: a polynomial is solvable by radicals if and only if its Galois group is a solvable group, explaining the lack of a general formula for quintic and higher-degree equations.
• Mastery requires careful attention to the definitions of normality and separability, and a disciplined approach to applying the inclusion-reversing correspondence of the Fundamental Theorem.