Timeline for Any other definition for algebraic number than the root of algebraic equation?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 29, 2014 at 12:17 | comment | added | XL _At_Here_There | Joel, your answer or the definition by model theory is so universal, I am wondering whether there is any linkage between transcendental number theory and such a definition. | |
| Apr 4, 2013 at 3:31 | vote | accept | XL _At_Here_There | ||
| Mar 8, 2013 at 1:47 | comment | added | Yemon Choi | Joel: stubby fingers have undone me again :) (It was meant to be "informative", as you no doubt guessed | |
| Mar 8, 2013 at 0:32 | vote | accept | XL _At_Here_There | ||
| Apr 4, 2013 at 3:31 | |||||
| Mar 8, 2013 at 0:31 | vote | accept | XL _At_Here_There | ||
| Mar 8, 2013 at 0:32 | |||||
| Mar 8, 2013 at 0:24 | comment | added | Joel David Hamkins | Oh, I'm very sorry if my answer has caused any looseness! :-) I probably agree with you, though, since I am unsure myself what kind of answer the OP may have had in mind. Meanwhile, I took the opportunity to mention a notion of algebraicity that I suspect is not so widely known outside logic, but which I find to be very interesting and easy to understand (and which answers the question). | |
| Mar 7, 2013 at 18:58 | comment | added | Yemon Choi | This is very unfirmative, but I can't help feeling it answers a different question from the one intended by the OP | |
| Mar 7, 2013 at 18:37 | comment | added | Joel David Hamkins | In any sufficiently saturated structure, an object is algebraic if and only if it has finite orbit under the action of the automorphism group. But these concepts are not equivalent in an arbitrary structure, for non-algebraic elements can nevertheless be fixed by every automorphism, as they are in $\langle\mathbb{R}, {+},{\cdot},0,1,\lt\rangle$. But meanwhile, an element is algebraic if and only if it has finite orbit in every elementary extension of the structure. | |
| Mar 7, 2013 at 17:15 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Improved exposition; added 167 characters in body
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| Mar 7, 2013 at 12:32 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |