Septic equation

Because they have an odd degree, septic functions appear similar to quintic and cubic functions when graphed, except they may possess additional local maxima and local minima (up to three maxima and three minima).

To give an example of an irreducible but solvable septic, one can generalize the solvable de Moivre quintic to get, where the auxiliary equation is This means that the septic is obtained by eliminating u and v between x = u + v, uv + α = 0 and u7 + v7 + β = 0.

The general septic equation can be solved with the alternating or symmetric Galois groups A7 or S7.

[1] 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.

[2] 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.

Graph of a polynomial of degree 7, with 7 real roots (crossings of the x axis) and 6 critical points . Depending on the number and vertical location of the minima and maxima , the septic could have 7, 5, 3, or 1 real root counted with their multiplicity; the number of complex non-real roots is 7 minus the number of real roots.
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