## 305 -Fields (4)

Suppose that ${mathbb F}$ is a field and that $Ssubset{mathbb F}.$ It may be that $S$ is also a field, using the same operations of ${mathbb F}.$ For example, if ${mathbb F}={mathbb R},$ then we could have $S={mathbb Q}.$

Definition 15. If ${mathbb F}$ is a field and $Ssubset{mathbb F},$ we say that $S$ is a subfield of ${mathbb F}$ if $S$ is a field with the operations of ${mathbb F}.$

Let’s examine this definition in some detail. Part of what this is saying is that

1. If $a,bin S,$ then also $a+bin S,$ i.e., $S$ is closed under addition.
2. If $a,bin S,$ then also $atimes bin S,$ i.e., $S$ is closed under multiplication.

However, this is not enough. For example, ${mathbb N}$ is not a field but it is closed under the addition and multiplication operations of ${mathbb R}.$  The problem with ${mathbb N}$ is that it does not have additive or multiplicative inverses of its elements.

Proposition 16. Suppose that ${mathbb F}$ is a field and that $a,bin{mathbb F}.$

•  If $a+b=a$ then $a=0.$
• If $ane0$ and $ab=a$ then $b=1.$

Proof. Add the additive inverse $-a$ to both sides of the first equation, and multiply by the multiplicative inverse $a^{-1}$ both sides of the second equation. ${sf QED}$

The point of Proposition 16 is the following: Suppose that $S$ is a subfield of ${mathbb F}.$ Write $0^{mathbb F}$ for the $0$-th element of ${mathbb F}$ and $0^S$ for the $0$-th element of $S.$ Then $0^S=0^{mathbb F},$ in particular, $0$ must belong to $S.$ Similarly, $1^S=1^{mathbb F},$ so $1$ belongs to $S,$ as long as $S$ contains some element other than $0.$ But, of course, if $S$ is to be a field, then it must have at least two elements, so one of them must be different from $0.$

Proposition 17. Suppose ${mathbb F}$ is a field and that $a,bin{mathbb F}.$

• If $a+b=0$ then $b=-a.$
• If $ab=1$ then $ane0$ and $b=a^{-1}.$ $Box$

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 $S$ is a subfield of ${mathbb F},$ and $ain S$ then the additive inverse of $a$ from the point of view of ${mathbb F}$ and its additive inverse from the point of view of $S$ must coincide. Similarly, the multiplicative inverse from the point of view of $S$ of any nonzero element of $S$ is the same as its multiplicative inverse from the point of view of ${mathbb F}.$ Hence, to properties 1,2 listed above we can add:

3. If $ain S,$ then $-ain S.$

And:

4. If $ain S,$ and $ane0,$ then $a^{-1}in S.$

It turns out that 1–4 characterize subfields:

Theorem 18. Suppose ${mathbb F}$ is a field and $Ssubseteq {mathbb F}.$ If $S$ satisfies 1–4 and has at least two elements, then $S$ is a subfield of ${mathbb F}.$

Noticed that we cannot remove the assumption that $S$ has two elements. For example, $S={0}$ 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.