Fields and Field Extensions: Abstract Algebra Intro Guide

A field extension K/F is a field K containing a smaller field F, like ℚ ⊆ ℝ or ℝ ⊆ ℂ. The degree [K : F] is the dimension of K as a vector space over F — how much bigger K is. ℂ over ℝ has degree 2 (basis {1, i}); ℝ over ℚ has infinite degree.

Extensions grow number systems by adjoining roots: ℚ(√2) is ℚ with a square root of 2 attached. Finite fields, algebraic number theory, and Galois theory are all towers of such extensions, with degrees multiplying along the way.

Key Concepts

• Notation: K/F means F ⊆ K (not a quotient).
• Degree: [K : F] = dim_F(K), finite or infinite.
• Simple extension: F(α), the smallest field containing F and α.
• Minimal polynomial: the monic polynomial over F of least degree having α as a root.
• Tower law: [K : F] = [K : E]·[E : F] for F ⊆ E ⊆ K.
• Adjoining a root: if p(x) is irreducible over F of degree n with root α, then [F(α) : F] = n, basis {1, α, …, α^{n−1}}.

Formulas

• Tower law: [K : F] = [K : E] · [E : F] for F ⊆ E ⊆ K. (Dimensions; no physical units.)
• Simple extension degree: [F(α) : F] = deg(minimal polynomial of α).
• Examples: [ℚ(√2) : ℚ] = 2; [ℂ : ℝ] = 2; [ℚ(∛2) : ℚ] = 3.
• Limit: degrees multiply only along chains of subfields.

Worked Example

The minimal polynomial of √2 over ℚ is x² − 2 (irreducible by Eisenstein, p = 2), so [ℚ(√2) : ℚ] = 2 with basis {1, √2}; every element is uniquely a + b√2. Multiply (3 + 2√2)(1 − √2) using (√2)² = 2: 3·1 = 3; 3·(−√2) = −3√2; 2√2·1 = 2√2; 2√2·(−√2) = −4. Sum: 3 − 3√2 + 2√2 − 4 = −1 − √2 — still of the form a + b√2. Tower: [ℚ(√2, √3) : ℚ] = 2·2 = 4, basis {1, √2, √3, √6}.

Common Misconceptions

• "K/F means K divided by F." The slash is not a quotient — it is just notation for "K over F," a field containing F.
• "Adjoining √2 to ℚ gives {a + b√2} with a, b integers." Coefficients come from the base field ℚ, not ℤ — otherwise you would not get a field (no inverses).
• "Every extension has finite degree." ℝ over ℚ is infinite-dimensional; finiteness is a special, powerful property, not the default.

Real-World Applications

• Galois theory: splitting fields of polynomials are towers of simple extensions; their degrees bound Galois group sizes.
• Finite fields: GF(p^n) is a degree-n extension of ℤ/pℤ — the fields AES encryption computes in.
• Constructibility: a length is ruler-and-compass constructible iff it lies in a tower of degree-2 extensions over ℚ — proving angle trisection impossible.
• Coding theory: Reed–Solomon codes live in extension fields GF(2^n), built by adjoining roots of irreducible polynomials.

A field extension grows a field by adjoining new elements; the degree measures the growth, the tower law multiplies degrees along chains, and irreducible polynomials are the seeds every finite extension grows from.