Is there an upper bound on the cardinality of a field?
The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than continuum?
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$\begingroup$ Have you heard of a transcendence basis? $\endgroup$– anonJul 15, 2014 at 21:32
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$\begingroup$ No, I have not... $\endgroup$– Marco FloresJul 16, 2014 at 3:11
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3 Answers
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Yes, There are arbitrarily large fields. This follows from a result in logic, namely the Upward Löwenheim–Skolem theorem (see http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem).
To be more concrete, consider the countable algebriacly closed field, $\bar{\mathbb{Q}}$. Every element of this field is algebraic (meaning it is a root of a polynomial). However, we can begin adding "transcendental" elements, like $\pi$ and $e$. These don't affect the "field properties" of the field. Then, we can add any cardinality of transcendentals to construct a field of that particular size.
answered Jul 15, 2014 at 21:32
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$\begingroup$ Well there are only at most $\beth_1$ transcendentals to add $\endgroup$ Jul 15, 2014 at 21:45
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4$\begingroup$ @John: The term "Transcendental" is not taken in the context of real/complex numbers here. It is taken in the context of adding elements which are not realizations of types over the model. Or in simpler words, not algebraic over a given field. $\endgroup$– Asaf Karagila ♦Jul 15, 2014 at 21:50
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Given any set $X$, you can come up with an integral domain $\mathbb Q[X]$, of polynomials with each $x\in X$ an indeterminate. This ring is an integral domain, so it has a field of fractions, and that field of fractions has at least $X$ elements.
If $X$ is infinite, I believe the field of fractions will have the same cardinality as $X$, but don't quote me.
answered Jul 15, 2014 at 21:33
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1$\begingroup$ It will be, at least if we assume the axiom of choice, which I assume we are. :-) $\endgroup$– Asaf Karagila ♦Jul 15, 2014 at 21:53
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No, surreal numbers form a 'field' which is a proper class, it contains all ordinals as a subclass for example. It also has subfields of every cardinality, see http://en.wikipedia.org/wiki/Surreal_number.
