Suppose that is a field and that It may be that is also a field, using the same operations of For example, if then we could have
Definition 15. If is a field and we say that is a subfield of if is a field with the operations of
Let’s examine this definition in some detail. Part of what this is saying is that
- If then also i.e., is closed under addition.
- If then also i.e., is closed under multiplication.
However, this is not enough. For example, is not a field but it is closed under the addition and multiplication operations of The problem with is that it does not have additive or multiplicative inverses of its elements.
Proposition 16. Suppose that is a field and that
- If then
- If and then
Proof. Add the additive inverse to both sides of the first equation, and multiply by the multiplicative inverse both sides of the second equation.
The point of Proposition 16 is the following: Suppose that is a subfield of Write for the -th element of and for the -th element of Then in particular, must belong to Similarly, so belongs to as long as contains some element other than But, of course, if is to be a field, then it must have at least two elements, so one of them must be different from
Proposition 17. Suppose is a field and that
- If then
- If then and
Proposition 17 can be proved by a very similar argument to that of Proposition 16, so I omit the proof. The point of this proposition is that if is a subfield of and then the additive inverse of from the point of view of and its additive inverse from the point of view of must coincide. Similarly, the multiplicative inverse from the point of view of of any nonzero element of is the same as its multiplicative inverse from the point of view of Hence, to properties 1,2 listed above we can add:
3. If then
4. If and then
It turns out that 1–4 characterize subfields:
Theorem 18. Suppose is a field and If satisfies 1–4 and has at least two elements, then is a subfield of
Noticed that we cannot remove the assumption that has two elements. For example, satisfies properties 1–4 but is not a field.
We will prove this theorem next lecture and use it to produce many new examples of fields.