## 305 -Fields (4)

Suppose that ${\mathbb F}$ is a field and that $S\subset{\mathbb F}.$ 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 $S\subset{\mathbb F},$ 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 $a,b\in S,$ then also $a+b\in S,$ i.e., $S$ is closed under addition.
2. If $a,b\in S,$ then also $a\times b\in S,$ 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 $a,b\in{\mathbb F}.$

•  If $a+b=a$ then $a=0.$
• If $a\ne0$ 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 $a,b\in{\mathbb F}.$

• If $a+b=0$ then $b=-a.$
• If $ab=1$ then $a\ne0$ 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 $a\in S$ 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 $a\in S,$ then $-a\in S.$

And:

4. If $a\in S,$ and $a\ne0,$ then $a^{-1}\in S.$

It turns out that 1–4 characterize subfields:

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

Notice 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.

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1. […] -Fields (5) At the end of last lecture we stated a theorem giving an easy characterization of subfields of a given field We begin by […]

2. […] the end of last lecture we stated a theorem giving an easy characterization of subfields of a given field We begin by […]