The bookUnit 2 · Rings and Fields47 / 83

Field extensions, splitting fields, finite fields

Why this is asked: Tower law plus 'degree = degree of the minimal polynomial' answers most questions. Know the standard degrees: [ℚ(∛2, ω) : ℚ] = 6, [ℚ(√2, √3) : ℚ] = 4.

Field extensions, degrees and splitting fields

The written notes for this page come with the Notes pack. The video, the visual and the practice below are free.

See pricing

See it move

Degrees multiply — computing [K:ℚ] and spotting the primitive elementinteractive

The tower law in practice, plus the subfield rule for finite fields that decides many options.

Unlock interactive visuals

The trap here

“An algebraic extension is a finite extension” — false

The field of all algebraic numbers Q̄ over $\mathbb{Q}$

Every element is algebraic, but the extension has infinite degree (it contains $\mathbb{Q}(2^{1/n})$ for every n).