15/12/19 15:25:52.68 VtRJxPeF.net
>>282
これもご参考。”J. Shipman showed in 2007 ”か。こんな定理が、2007年ですか？
URLﾘﾝｸ(en.wikipedia.org)
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.
Polynomials of prime degree have roots
J. Shipman showed in 2007 that if every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F, thus F is algebraically closed.
Shipman, Joseph (2007), "Improving the Fundamental Theorem of Algebra", Mathematical Intelligencer 29 (4), pp. 9?14, doi:10.1007/BF02986170, ISSN 0343-6993
URLﾘﾝｸ(dx.doi.org)
URLﾘﾝｸ(mathoverflow.net)
25 edited Nov 7 '13 at 6:10
A recent and very important contribution to the literature on the fundamental theorem of algebra is Joe Shipman's article
"Improving the Fundamental Theorem of Algebra," Math. Intelligencer 29 (2007), 9-14, doi:10.1007/BF02986170. Here is one of his results:
A field with the property that every polynomial whose degree is a prime number has a root is algebraically closed. This result is sharp in the sense that if any prime is omitted then the conclusion is false.
Shipman's paper should go a long way towards addressing Andrew L's question of whether there is a "purely algebraic proof" of the FTA.
The above result of Shipman's shows that we can limit the topology/analysis to proving that every polynomial over C of prime degree has a root; the rest is pure algebra