Def/Subfield



=Intersections of Subfields= Suppose that $X$ is a set of subfields of a field $F$. Then $\bigcap X$ is a subfield of $F$.

Since the intersection of subfields of $F$ is a subfield of $F$, we find that there exists a minimal subfield of $F$ (with respect to the order given by inclusion). This is called the defines::characteristic subfield of $F$.

=Field extensions as vector spaces= Suppose that $K/F$ is a field extension, i.e., $K$ is a field, and $F$ is a subfield of $K$. Then, $K$ has a natural structure of a $F$ vector space, given as follows:
 * Addition, subtraction, and zero in $K$ are given by the its addition as a field.
 * Scaling by $F$ is given by multiplication in $K$ (since $F$ is a subfield of $K$, the product of an element of $F$ with an element of $K$ is an element of $K$).

The defines::degree of the field extension $K/F$ is defined to be the dimension of $K$, viewed as an $F$ vector space. The degree of the field extensions $K/F$ is denoted by $[K:F]$.

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