Field extension

If $K$ and $L$ are fields and $K\subseteq L$ , then $L/K$ is said to be a field extension. We sometimes say that $L$ is a field extension of $K$ .

If $L/K$ is a field extension, then $L$ may be thought of as a vector space over $K$ . The dimension of this vector space is called the degree of the extension, and is denoted by $[L:K]$ .

Given three fields $K\subseteq L\subseteq M$ , then, if the degrees of the extensions $M/L$ , $L/K$ and $M/K$ , are finite, then are related by the tower law: $[M:K] = [M:L]\cdot[L:M]$

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Field extension