Minimal polynomials: the first relation determines the dimension

Let α be algebraic over F. Choose a nonzero polynomial of least degree vanishing at α, and divide by its leading coefficient to make it monic. Its degree is positive because a nonzero constant cannot vanish. If this polynomial factored into two lower positive-degree polynomials over F, evaluating at α would make their product zero in a field. One factor would vanish, contradicting minimality. Thus the minimal polynomial is irreducible over F.

Polynomial division proves both uniqueness and a useful divisibility test. Divide any polynomial g vanishing at α by the minimal polynomial m. The remainder has smaller degree and also vanishes at α, so it must be zero. Therefore m divides every relation. Two monic minimal relations divide each other and consequently agree. Notice the phrase over F: X² − 2 is minimal for √2 over Q, whereas X − √2 is minimal once √2 is already allowed as a coefficient.

If m has degree d, dividing by m reduces every polynomial expression to degree below d. Evaluation identifies F[α] with the quotient F[X]/(m). Since m is irreducible, this quotient is a field, so F[α] already equals F(α). Explicitly, for a polynomial h with h(α) nonzero, the greatest common divisor of h and m is one. A polynomial Bézout identity then produces a polynomial representing the inverse of h(α).

The expressions 1, α, up to α raised to d − 1 span the extension. They are also independent: any nontrivial linear dependence would be a lower-degree polynomial relation. Hence they form a basis over F. The extension degree [F(α):F] is defined as this vector-space dimension. It counts independent coordinates, not the number of field elements and not automatically the degree of the first polynomial someone happens to write down.

For the real cube root α of 2, the polynomial X³ − 2 is irreducible over Q by Eisenstein at the prime 2. Thus every element has a unique expression a + bα + cα² with rational coefficients. Multiplication is ordinary expansion followed by replacing α³ with 2. Uniqueness depends on irreducibility, not merely on the fact that the cubic vanishes at α. A reducible relation would give a spanning list that could still contain unnecessary coordinates.

Minimal-polynomial computations therefore have two obligations: show that the proposed polynomial vanishes and show it is irreducible over the stated field. Numerical approximation verifies neither obligation exactly. A decimal close to a root could instead be a nearby rational number, whose minimal polynomial is linear. When the base field changes, repeat the irreducibility argument; a polynomial irreducible over Q can factor after adjoining another root or adding new coefficients.