In algebra, a septic equation is an equation of the form
a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h = 0 , {\displaystyle ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h=0,\,}
where a ≠ 0. A septic function is a function of the form
f ( x ) = a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h {\displaystyle f(x)=ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h\,}
where a ≠ 0. In other words, it is a polynomial of degree seven. If a = 0, then f is a sextic function (b ≠ 0), quintic function (b = 0, c ≠ 0), etc. The equation may be obtained from the function by setting f(x) = 0. The coefficients a, b, c, d, e, f, g, h may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field or ring. 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). The derivative of a septic function is a sextic function, except possibly when the characteristic of the ring containing the coefficients is divisible by 7.
Solvable septics Some seventh degree equations can be solved by factorizing into radicals, but other septics cannot. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory. To give an example of an irreducible but solvable septic, one can generalize the solvable de Moivre quintic to get,
x 7 + 7 α x 5 + 14 α 2 x 3 + 7 α 3 x + β = 0 {\displaystyle x^{7}+7\alpha x^{5}+14\alpha ^{2}x^{3}+7\alpha ^{3}x+\beta =0\,} , where the auxiliary equation is
y 2 + β y − α 7 = 0 {\displaystyle y^{2}+\beta y-\alpha ^{7}=0\,} . This means that the septic is obtained by eliminating u and v between x = u + v, uv + α = 0 and u7 + v7 + β = 0. It follows that the septic's seven roots are given by
x k = ω k y 1 7 + ω k 6 y 2 7 {\displaystyle x_{k}=\omega _{k}{\sqrt[{7}]{y_{1}}}+\omega _{k}^{6}{\sqrt[{7}]{y_{2}}}}
where ωk is any of the 7 seventh roots of unity. The Galois group of this septic is the maximal solvable group of order 42. This is easily generalized to any other degrees k, not necessarily prime. Another solvable family is,
x 7 − 2 x 6 + ( α + 1 ) x 5 + ( α − 1 ) x 4 − α x 3 − ( α + 5 ) x 2 − 6 x − 4 = 0 {\displaystyle x^{7}-2x^{6}+(\alpha +1)x^{5}+(\alpha -1)x^{4}-\alpha x^{3}-(\alpha +5)x^{2}-6x-4=0\,}
whose members appear in Kluner's Database of Number Fields. Its discriminant is
Δ = − 4 4 ( 4 α 3 + 99 α 2 − 34 α + 467 ) 3 {\displaystyle \Delta =-4^{4}\left(4\alpha ^{3}+99\alpha ^{2}-34\alpha +467\right)^{3}\,}
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