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.

Advertisements

One Response to 305 -Fields (4)

  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 […]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: