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Field Theory & Galois Basics cheat sheet - grade college

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Math Grade college

Field Theory & Galois Basics Cheat Sheet

A printable reference covering field extensions, algebraic elements, splitting fields, Galois groups, and the fundamental theorem of Galois theory for college.

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Field theory studies how one field can sit inside a larger field and how polynomial roots behave inside those extensions. This cheat sheet helps students organize the basic language of extensions, algebraic elements, splitting fields, and automorphism groups. These ideas are essential for understanding why some polynomial equations can be solved by radicals and how symmetry controls roots.

The core tools are extension degrees, minimal polynomials, splitting fields, and field automorphisms that fix a base field. A finite extension L/KL/K is Galois when it is both normal and separable, and then its symmetries form the group Gal⁡(L/K)\operatorname{Gal}(L/K). The fundamental theorem of Galois theory gives an inclusion-reversing correspondence between intermediate fields and subgroups of the Galois group.

Key Facts

  • A field extension L/KL/K means K⊆LK \subseteq L and the operations on KK agree with the operations inherited from LL.
  • The degree of a finite extension is [L:K]=dim⁡KL[L:K]=\dim_K L, the dimension of LL as a vector space over KK.
  • If K⊆F⊆LK \subseteq F \subseteq L and the degrees are finite, then [L:K]=[L:F][F:K][L:K]=[L:F][F:K].
  • An element α∈L\alpha \in L is algebraic over KK if there is a nonzero polynomial f(x)∈K[x]f(x) \in K[x] such that f(α)=0f(\alpha)=0.
  • The minimal polynomial mα,K(x)m_{\alpha,K}(x) is the unique monic irreducible polynomial in K[x]K[x] with mα,K(α)=0m_{\alpha,K}(\alpha)=0.
  • A splitting field of f(x)∈K[x]f(x) \in K[x] is the smallest extension L/KL/K in which f(x)f(x) factors completely into linear factors.
  • The Galois group is Gal⁡(L/K)={σ∈Aut⁡(L):σ(a)=a for all a∈K}\operatorname{Gal}(L/K)=\{\sigma \in \operatorname{Aut}(L):\sigma(a)=a \text{ for all } a \in K\}.
  • For a finite Galois extension L/KL/K, the fundamental theorem gives F↔Gal⁡(L/F)F \leftrightarrow \operatorname{Gal}(L/F) between intermediate fields K⊆F⊆LK \subseteq F \subseteq L and subgroups of Gal⁡(L/K)\operatorname{Gal}(L/K).

Vocabulary

Field extension
A field extension L/KL/K is a pair of fields with K⊆LK \subseteq L, so elements of LL can be studied using scalars from KK.
Algebraic element
An element α\alpha is algebraic over KK if it satisfies some nonzero polynomial equation f(α)=0f(\alpha)=0 with f(x)∈K[x]f(x) \in K[x].
Minimal polynomial
The minimal polynomial of α\alpha over KK is the monic irreducible polynomial mα,K(x)∈K[x]m_{\alpha,K}(x) \in K[x] of least degree with root α\alpha.
Splitting field
A splitting field for f(x)f(x) over KK is the smallest field extension of KK containing all roots of f(x)f(x).
Galois group
The Galois group Gal⁡(L/K)\operatorname{Gal}(L/K) is the group of field automorphisms of LL that fix every element of KK.
Normal extension
An algebraic extension L/KL/K is normal if every irreducible polynomial in K[x]K[x] with one root in LL splits completely over LL.

Common Mistakes to Avoid

  • Confusing an extension with a quotient is wrong because L/KL/K in field theory usually means containment K⊆LK \subseteq L, not division or a quotient field.
  • Assuming every algebraic extension is Galois is wrong because a finite extension must also be normal and separable to be Galois.
  • Forgetting that automorphisms in Gal⁡(L/K)\operatorname{Gal}(L/K) fix KK is wrong because these maps may move elements of LL, but every a∈Ka \in K must satisfy σ(a)=a\sigma(a)=a.
  • Calling any field containing some roots a splitting field is wrong because the splitting field must contain all roots and be the smallest such extension over the base field.
  • Reversing the fundamental theorem correspondence is wrong because larger intermediate fields correspond to smaller subgroups, so the correspondence is inclusion-reversing.

Practice Questions

  1. 1 Find [Q(2,3):Q][\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}] and give a basis over Q\mathbb{Q}.
  2. 2 Find the minimal polynomial of 23\sqrt[3]{2} over Q\mathbb{Q} and compute [Q(23):Q][\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}].
  3. 3 Let f(x)=x2−5f(x)=x^2-5 over Q\mathbb{Q}. Identify its splitting field and describe Gal⁡(L/Q)\operatorname{Gal}(L/\mathbb{Q}).
  4. 4 Explain why the splitting field of x3−2x^3-2 over Q\mathbb{Q} is larger than Q(23)\mathbb{Q}(\sqrt[3]{2}).

Understanding Field Theory & Galois Basics

A useful way to build an extension is to start with one new algebraic number and include every expression that field rules require. If a number has minimal polynomial of degree two, then expressions in that number can usually be reduced to a form using just two basis pieces. For example, after adjoining the square root of two to the rational numbers, every element has the form rational number plus rational number times the square root of two.

Higher degree minimal polynomials create longer bases. This reduction process is why polynomial division and linear algebra are central tools in field theory.

The roots of a minimal polynomial are called conjugates. They show what an automorphism may do to a chosen element. An automorphism must preserve addition, multiplication, and every element of the base field.

Therefore it cannot send a root to an arbitrary number. It must send it to another root of the same minimal polynomial. In the square root of two example, there are two possible images, the positive square root and the negative square root.

This simple swap is the first example of root symmetry. For more complicated polynomials, the allowed root permutations can be much more restricted than all possible permutations.

Two conditions explain when an extension has enough symmetry for the main Galois correspondence to work cleanly. Normality means that when a polynomial over the base field has one relevant root in the extension, all of its conjugate roots are present too. Separability means that roots do not repeat.

Over the rational numbers and other fields of characteristic zero, irreducible polynomials are automatically separable. Repeated roots become an important issue in fields whose characteristic is a prime number. Students should learn to check the derivative of a polynomial in that setting, since a common factor with its derivative signals repeated roots.

The subgroup and intermediate field correspondence reverses direction for a clear reason. More field elements give automorphisms more things to fix, so fewer automorphisms remain. Conversely, a larger subgroup imposes more fixed conditions, leaving a smaller fixed field.

This is not just a counting trick. It lets algebra problems become group problems.

A field built by repeatedly taking roots corresponds to a group with a special layered structure. That connection explains the classical result that a general polynomial of degree five cannot be solved using radicals.

These ideas first appear in abstract algebra courses, but they continue into number theory, geometry, coding theory, and cryptography through finite fields. The practical applications usually use finite fields rather than extensions of the rational numbers, yet the same habits matter. Keep track of the base field at every step.

A polynomial can be irreducible over one field but factor over a larger one. Distinguish one chosen root from the full splitting field containing every root. When working examples, list a basis, identify the conjugates, and test which root permutations preserve all algebraic relations.