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.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.