Embedding Fields in Matrix Rings

abstract-algebra field-theory extension-field

Let $F$ be a field and let $K$ be an extension of $F$ of degree $n$.

Then $\mu: K \to End_F(K)$ given by $\mu(a)(x)=ax$ is an injective ring homomorphism.

Choosing a basis for $K$ over $F$ gives a ring isomorphism $End_F(K) \cong M_n(F)$ and so an embedding $K \to M_n(F)$.

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