Extension $K$ of field $F$ forms a vector space over $F$

Question

The text am reading says that the following can be seen 'easily':

Suppose $K$ and $F$ are fields. Then $K$ is called an extension of $F$ if $F \subseteq K$. It can be seen 'easily' that $K$ forms a vector space over $F$.

But am rather new to this type of algebra, and I couldn't see the above easily .... Could I have some simple explanations to explain why or perhaps some examples ?

Answer 1

It is easy to verify that $K$ satisfies the axioms for a vector space over $F$, since these axioms are essentially just the field axioms cast in a different context. For example, if

$a \in F \tag 1$

and

$b, c \in K, \tag 2$

then

$a(b + c) = ab + ac, \tag 3$

i.e., multiplication by elements of $F$ distributes over addition in $K$; likewise, with

$a, b \in F \tag 3$

and

$c \in K, \tag 4$

we have

$(ab)c = a(bc), \tag 5$

the usual formulation of the associativity of scalar-vector multiplication; the other vector space axioms are also simple consequences of their field-theoretic counterparts.

A concrete instance may be had by taking

$F = \Bbb Q, \tag 6$

the rationals, and

$K = \Bbb R, \tag 7$

the reals. The reader is undoubtedly familiar with the (obvious) arithmetic in this case.