Septic equation
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The general septic equation can be solved with the [[alternating group|alternating]] or [[symmetric group|symmetric]] [[Galois group]]s {{math|''A''7}} or {{math|''S''7}}.{{Cite book |last=King |first=R. Bruce |date=2009 |title=Beyond the Quartic Equation |url= https://books.google.com/books?id=9cKX_9zkeg4C&q=septic+equation&pg=PA143 |location=Boston/Basel/Berlin |publisher=Birkhäuser |page=143, 144 |isbn=978-0-8176-4849-7}} Such equations require [[hyperelliptic function]]s and associated [[theta function]]s of [[genus (mathematics)|genus]] 3 for their solution. However, these equations were not studied specifically by the nineteenth-century mathematicians studying the solutions of algebraic equations, because the [[sextic equation]]s' solutions were already at the limits of their computational abilities without computers. |
The general septic equation can be solved with the [[alternating group|alternating]] or [[symmetric group|symmetric]] [[Galois group]]s {{math|''A''7}} or {{math|''S''7}}.{{Cite book |last=King |first=R. Bruce |date=2009 |title=Beyond the Quartic Equation |url= https://books.google.com/books?id=9cKX_9zkeg4C&q=septic+equation&pg=PA143 |location=Boston/Basel/Berlin |publisher=Birkhäuser |page=143, 144 |isbn=978-0-8176-4849-7}} Such equations require [[hyperelliptic function]]s and associated [[theta function]]s of [[genus (mathematics)|genus]] 3 for their solution. However, these equations were not studied specifically by the nineteenth-century mathematicians studying the solutions of algebraic equations, because the [[sextic equation]]s' solutions were already at the limits of their computational abilities without computers. |
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Septics are the lowest order equations for which it is not obvious that their solutions may be obtained by composing ''continuous functions'' of two variables. [[Hilbert's thirteenth problem|Hilbert's 13th problem]] was the conjecture this was not possible in the general case for seventh-degree equations. [[Vladimir Arnold]] solved this in 1957, demonstrating that this was always possible.{{Cite book |
Septics are the lowest order equations for which it is not obvious that their solutions may be obtained by composing ''continuous functions'' of two variables. [[Hilbert's thirteenth problem|Hilbert's 13th problem]] was the conjecture this was not possible in the general case for seventh-degree equations. [[Vladimir Arnold]] solved this in 1957, demonstrating that this was always possible.{{Cite book |first=Vasco |last=Brattka |date=2007 |chapter=From Hilbert’s 13th Problem to the theory of neural networks: constructive aspects of Kolmogorov’s Superposition Theorem |chapter-url=https://books.google.com/books?id=SpTv44Ia-J0C&pg=PA254 |title=Kolmogorov's Heritage in Mathematics |location=Berlin/Heidelberg |publisher=Springer |page=253–280 |isbn=978-3-540-36349-1}} However, Arnold himself considered the ''genuine'' Hilbert problem to be whether for septics their solutions may be obtained by superimposing ''algebraic functions'' of two variables.{{citation |url=http://www.pdmi.ras.ru/~arnsem/Arnold/arnlect1.ps.gz |title=From Hilbert's Superposition Problem to Dynamical Systems |author=V.I. Arnold |page=4}} As of 2023, the problem is still open. |
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==Galois groups== |
==Galois groups== |
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