Field Extensions

[K:F] = _F K; degrees multiply in towers.

The idea

A field extension is a pair of fields $F \subseteq K$, and its degree measures how much larger $K$ is than $F$.

The larger field carries two structures at once. It is a field in its own right, and it is a vector space over $F$: elements of $K$ can be added, and they can be scaled by elements of $F$, and the vector space axioms follow from the field axioms of $K$. So $K$ has a dimension over $F$, and that dimension is the degree, $[K : F] = \dim_F K.$ For example, $\mathbb{C}$ has basis $1, i$ over $\mathbb{R}$, so $[\mathbb{C} : \mathbb{R}] = 2$. The degree is often finite even when both fields are infinite, which is what makes it a usable measure of size.

The tower law. If $F \subseteq K \subseteq L$, then $[L : F] = [L : K]\,[K : F],$ because the products of a basis of $L$ over $K$ with a basis of $K$ over $F$ form a basis of $L$ over $F$. Adjoining a root of an irreducible polynomial of degree $n$ produces an extension of degree $n$, so we can measure an extension built in stages by multiplying the degrees of the stages.

Ways to work on it

• Walkthrough. Degree as a dimension and the tower law.
• Practice. Apply the tower law.
• Hardest. Find the degree of an extension generated by two square roots.

Not sure where to start? Take the ten-question placement test.