Field extensions

Objective. Calculate extension degrees and minimal polynomials.

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2. Worked example
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Learn: the key idea

A field extension K/F is formed when F is a subfield of K, and the degree [K:F] is the dimension of K as a vector space over F; if α is algebraic over F, the minimal polynomial of α is the monic polynomial of least degree in F[x] with α as a root, and [F(α):F] equals the degree of that minimal polynomial. The tower law states [K:F] = [K:E][E:F] for intermediate field E, allowing degrees to be computed by multiplying through a chain of extensions. For example, ℚ(√2) has degree 2 over ℚ since the minimal polynomial of √2 is x²−2, while ℚ(√2,√3) has degree 4 over ℚ since √3 is not in ℚ(√2).

Worked example

Find the degree of the field extension ℚ(∛2) over ℚ.

1 $The minimal polynomial of ∛2 over ℚ is x³-2$
2 $Check x³-2 is irreducible over ℚ by rational root theorem: no rational root satisfies it$
3 $Since it is irreducible of degree 3, it is the minimal polynomial$
4 $The degree of the extension [ℚ(∛2):ℚ] equals 3$

Answer: 3

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Common mistake. Students assume degrees of extensions always add rather than multiply through towers; for ℚ⊂ℚ(√2)⊂ℚ(√2,√3), the correct total degree is 2×2=4, not 2+2=4 (which happens to coincide numerically here but is conceptually wrong, as seen with cubic extensions where addition and multiplication clearly differ).

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