## Rocky Mountain Journal of Mathematics

### Realizing infinite cardinal numbers via maximal chains of intermediate fields

#### Abstract

For each nonzero cardinal number $\kappa$, there is a field extension $\mathbf{Q} \subseteq L$ and a maximal chain $\mathcal{H}_{\kappa}$ of intermediate fields going from $\mathbf{Q}$ to $L$ such that the cardinal number of $\mathcal{H}_{\kappa}$ is $\kappa$. If $\kappa$ is infinite, then for all infinite cardinals $\nu%_1 \lt \nu_2 \leq \lt \kappa$, it can be arranged that $\mathcal{H}_{\nu%_1 } \subset %\mathcal{H}_{\nu_2} \subseteq \mathcal{H}_{\kappa}$. However, there exists an infinite cardinal for which there does not exist a field extension $L/K$ such that $\kappa$ is the supremum of the cardinalities of chains of intermediate fields going from $K$ to $L$. For a field extension $L/K$, this supremum of cardinalities has been denoted $\lambda(L/K)$. If $L/K$ is infinitely generated, we reduce its calculation to set theory, as follows. Let $\ala$ be the infimum of the cardinalities of generating sets of $L/K$. Let $\Omega(\aleph_{\alpha})$ be the supremum of the cardinalities of chains %that consist of subsets of a set of cardinality $\ala$. ($\Omega(\aleph_{\alpha})$ is equal to what has been called $\mbox{ded\,}(\aleph_{\alpha})$ in the literature.) Then $\ala\lt \oa\leq \p$; and if $\alpha > 0$ (but not necessarily if $\alpha =0$), then $\lambda(L/K)=\Omega(\aleph_{\alpha})$.

#### Article information

Source
Rocky Mountain J. Math., Volume 44, Number 5 (2014), 1471-1503.

Dates
First available in Project Euclid: 1 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1420071551

Digital Object Identifier
doi:10.1216/RMJ-2014-44-5-1471

Mathematical Reviews number (MathSciNet)
MR3295639

Zentralblatt MATH identifier
1332.12009

#### Citation

Dobbs, David E.; Heitmann, Raymond C. Realizing infinite cardinal numbers via maximal chains of intermediate fields. Rocky Mountain J. Math. 44 (2014), no. 5, 1471--1503. doi:10.1216/RMJ-2014-44-5-1471. https://projecteuclid.org/euclid.rmjm/1420071551

#### References

• J. Barwise, Handbook of mathematical logic (Studies in logic and the foundations of mathematics), Elsevier, Amsterdam, 1977.
• E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form $D+M$, Michigan Math. J. 20 (1973), 79–95.
• D.E. Dobbs, On chains of overrings of an integral domain, Int. J. Comm. Rings 1 (2002), 173–179.
• ––––, On infinite chains of intermediate fields, Inter. Elect. J. Alg. 11 (2012), 165–176.
• D.E. Dobbs and B. Mullins, On the lengths of maximal chains of intermediate fields in a field extension, Comm. Alg. 29 (2001), 4487–4507.
• R. Gilmer and W. Heinzer, Jónsson $\omega_0$-generated algebraic field extensions, Pacific J. Math. 128 (1987), 81–116.
• P.R. Halmos, Naive set theory, Van Nostrand, Princeton, 1960.
• T.W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
• N. Jacobson, Lectures in abstract algebra, Volume III, Theory of fields and Galois theory, Van Nostrand, Princeton, 1964.
• M.M. Zuckerman, Sets and transfinite numbers, Macmillan, New York, 1974.