vector space over a field

A vector space is a collection of objects you can add together and scale up or down. The “scaling” is done by numbers drawn from a fixed field, called scalars. Think of arrows in the plane: you can add two arrows tip-to-tail, and you can stretch or shrink an arrow by any real number. The plane is a vector space over the real numbers.

Precisely, a vector space over a field K is a set V with an addition of vectors and a multiplication of vectors by scalars from K, satisfying natural rules (addition is commutative and associative with a zero vector and negatives; scaling distributes over addition and is compatible with field multiplication). Every such space has a dimension: the number of vectors in a basis, a minimal spanning set.

In Galois theory this is the measuring tape. Any field extension L/K can be viewed as a vector space over K: you can add elements of L and multiply them by elements of K. Its dimension as a K-vector space is the degree of the extension, written [L : K] — the precise number that quantifies how much bigger L is than K.