Field Extensions and Their Degrees
Adjoining a root to a field produces a larger field, which is a vector space over the smaller one. Its dimension is called the degree, and degrees multiply along a tower of extensions.
What a learner can do afterwards
- Compute the degree of a simple extension from a minimal polynomial
- Apply the tower law to a compound extension
- Explain why a ruler and compass cannot produce the cube root of two
1 · Read
You build an extension by adjoining a root to a base field, and the result is a vector space over the base. The dimension is the degree, and for a simple extension it equals the degree of the minimal polynomial: the monic irreducible polynomial the new element satisfies. Adjoining the square root of a prime to rationals uses a quadratic, so the degree is 2. Adjoining the real cube root of 2 uses x cubed minus 2, irreducible by Eisenstein, so the degree is 3.
You stack extensions with the tower law, which multiplies degrees. Adjoin one square root to get degree 2, show the second square root is still outside, and adjoin it for another factor of 2, giving total 4 for two distinct primes. Combine a degree 2 square root with a degree 3 cube root of coprime degree to get 6. Adjoining the imaginary unit uses x squared plus 1, so the degree is 2, and primitive 5th roots give degree 4.
You test ruler and compass builds the same way. Each compass step can only double the field, so constructible degrees are powers of 2. Doubling the cube needs the cube root of 2, of degree 3, which is not a power of 2, so you can tell at once that the construction is impossible.
You read each degree off its minimal polynomial, multiply degrees along towers, and rule out builds whose degree is not a power of 2.
More to read
3.6 Zeros of Polynomial Functions
Zeros of polynomials determine extension degrees. An irreducible polynomial of degree d adjoined to the base gives degree d. Quadratics give degree two, cubics give degree three. Eisenstein often proves irreducibility. Tower law then multiplies degrees for combined extensions.
Quadratic formula proof review
Quadratic formula proof review shows degree two behavior. The discriminant decides whether roots already lie in the base. Adjoining a missing square root doubles the field. Cube roots need degree three and behave differently. Ruler and compass allow only repeated doubling, hence powers of two.
3.5 Irrational Numbers
Irrational numbers provide the first extensions. Square roots of primes are irrational, giving degree two. Cube roots give degree three and block classical constructions. Doubling the cube needs degree three, not a power of two.
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