The minimal polynomial: a number's fingerprint
An algebraic number α over F may satisfy many polynomials. Among them there is a unique “best” one: the monic polynomial of lowest [[degree-of-a-polynomial|degree]] that has α as a root. This is the minimal polynomial of α over F.
Two facts make it special. First, it is irreducible — a prime polynomial over F, with no factorization into smaller-degree pieces. Second, it divides every polynomial that α satisfies. So it is genuinely the simplest equation α obeys.
Minimal polynomial of α = √2 + √3 over Q: α = √2 + √3 α^2 = 2 + 2√6 + 3 = 5 + 2√6 α^2 − 5 = 2√6 (α^2 − 5)^2 = 24 α^4 − 10α^2 + 25 = 24 α^4 − 10α^2 + 1 = 0 Minimal polynomial: x^4 − 10x^2 + 1 (degree 4) So [Q(√2 + √3) : Q] = 4.
Why the degree is the dimension
If the minimal polynomial of α has degree n, then F(α) has basis {1, α, α², …, αⁿ⁻¹}, and [F(α) : F] = n. The reason: any higher power of α can be rewritten using the minimal polynomial, just as α² = 2 let us collapse everything in Q(√2).
Splitting fields: room for all the roots
A polynomial like x² − 2 has no roots in Q, but inside Q(√2) it factors as (x − √2)(x + √2) — it splits into linear pieces. The smallest extension of F in which a given polynomial factors completely into linear factors is its splitting field.
Sometimes you must adjoin more than one root, and even non-real numbers. The splitting field of x³ − 2 over Q needs the real cube root ∛2 and a complex cube root of unity ω; it is Q(∛2, ω) with degree 6. Splitting fields are the natural homes where Galois theory lives — every root present, nothing extra.