Degree of a field extension

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Definition

Suppose $L/K$ is a field extension. Then $L$ can be viewed as a vector space over the field $K$.

The degree of the field extension $L/K$ is the dimension of that vector space.

A field extension of degree 2, 3, 4, 5, ... is called a quadratic, cubic, quartic, quintic, ... field extension.

Examples

• For any field $K$, the field extension $K/K$ is degree 1.

• $\mathbb {C} /\mathbb {R}$ is degree 2, since $\mathbb {C}$ can be viewed as a 2 dimensional vector space over $\mathbb {R}$, since every element of $\mathbb {C}$ can be written as $z=x+iy$ for $x,y\in \mathbb {R}$.

• $\mathbb {C} /\mathbb {Q}$ is an extension of infinite degree.

See also

• Finite field extension
• Infinite field extension