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

And:

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

Notice 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.

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A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

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I don't think you need too much in terms of prerequisites. An excellent reference is MR3616119. Tomkowicz, Grzegorz(PL-CEG2); Wagon, Stan(1-MACA-NDM). The Banach-Tarski paradox. Second edition. With a foreword by Jan Mycielski. Encyclopedia of Mathematics and its Applications, 163. Cambridge University Press, New York, 2016. xviii+348 pp. ISBN: 978-1-10 […]

For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.

[…] -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 […]

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