Timeline for Sum of subfields of $\mathbb{C}$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2018 at 11:15 | comment | added | YCor | It doesn't matter in my construction. So if you can produce such decomposition in any characteristic and unbounded cardinals (as it seems), it works. | |
| Dec 31, 2018 at 10:57 | comment | added | Jeremy Rickard | @YCor Ah,right! For some reason (unrelated to anything you actually wrote!) I thought you were trying to decompose a fixed subfield of $C$. So this deals with any algebraically closed field with infinite transcendence degree, I think. I don't think the characteristic matters? | |
| Dec 31, 2018 at 9:15 | comment | added | YCor | I should have written $C=A+B$ (not $A\oplus B$), with $A,B\neq C$, and pick $a\in A-B$, $b\in B-A$, and assume that the subfield contains $a,b$. I construct a subfield $K$ of $C$, stable under $p$, containing $a$ and $b$, and of any prescribed cardinal in $[\omega,|C|]$. So $K=(A\cap K)+(B\cap K)$ and both are proper subfields of $K$. | |
| Dec 31, 2018 at 9:08 | comment | added | Jeremy Rickard | @YCor I don’t think I can work out the details of your construction. In particular, how do you make sure that you end up with proper subfields? | |
| Dec 28, 2018 at 9:11 | comment | added | YCor | Here's a way to deduce alg. closed fields of char 0 and arbitrary infinite card. First, the above construction works for arbitrary uncountable regular card. Second, given a large alg. closed field $C=A\oplus B$, sum of two subfields, choose a (set-wise) projection $p$ onto $A$ such that $p(x)-x\in B$ for all $x$. Then the operations of taking the field generated by a subset, taking its relative alg. closure, and taking union with the proj. to $A$, are finitary and preserve infinite card, and hence every infinite subset is contained in a $p$-stable alg. closed subfield of the same cardinal. | |
| Dec 27, 2018 at 19:26 | history | answered | Jeremy Rickard | CC BY-SA 4.0 |