High Quality Content by WIKIPEDIA articles! The general septic equation can be solved with the alternating or symmetric Galois groups A7 or S7. Such equations require hyperelliptic functions and associated theta functions of 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 equations' solutions were already at the limits of their computational abilities...
High Quality Content by WIKIPEDIA articles! The general septic equation can be solved with the alternating or symmetric Galois groups A7 or S7. Such equations require hyperelliptic functions and associated theta functions of 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 equations' solutions were already at the limits of their computational abilities without computers. Septics are the lowest order equations for which it is not obvious that their solutions may be obtained by superimposing continuous functions of two variables. Hilbert's 13th problem was the conjecture this was not possible in the general case for seventh order equations. Vladimir Arnold solved this in 1957, demonstrating that this was always possible.
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