is a rational function of the coefficients, which when these are
replaced by their values as functions of the roots becomes, according
to the sign given to the quadric radical, = L^3 or M^3; taking it =
L^3, the cube root of the expression has the three values L, [omega]L,
[omega]^2 L; and LM divided by the same cube root has therefore the
values M, [omega]^2M, [omega]M; whence finally the expression
(1/3)[p + [root 3]{1/2(L^3 + M^3 + [root](L^3 - M^3)^2)} + LM /
[root 3]{1/2L^3 + M^3 + [root](L^3 - M^3)^2}]
has the three values
(1/3)(p + L + M), (1/3)(p + [omega]L + [omega]^2 M),
(1/3)(p + [omega]^2 L + [omega]M);
that is, these are = a, b, c respectively. If the value M^3 had been
taken instead of L^3, then the expression would have had the same
three values a, b, c. Comparing the solution given for the cubic x^3 +
qx - r = 0, it will readily be seen that the two solutions are
identical, and that the function r^2 - (4/27)q^3 under the radical
sign must (by aid of the relation p = 0 which subsists in this case)
reduce itself to (L^3 - M^3)^2; it is only by each radical being equal
to a rational function of the roots that the final expression _can_
become equal to the roots a, b, c respectively.
20. The formulae for the cubic were obtained by J.L. Lagrange
(1770-1771) from a different point of view. Upon examining and comparing
the principal known methods for the solution of algebraical equations,
he found that they all ultimately depended upon finding a "resolvent"
equation of which the root is a + [omega]b + [omega]^2 c + [omega]^3 d +
..., [omega] being an imaginary root of unity, of the same order as the
equation; e.g. for the cubic the root is a + [omega]b + [omega]^2 c,
[omega] an imaginary cube root of unity. Evidently the method gives for
L^3 a quadric equation, which is the "resolvent" equation in this
particular case.
For a quartic the formulae present themselves in a somewhat different
form, by reason that 4 is not a prime number. Attempting to apply it to
a quintic, we seek for the equation of which the root is (a + [omega]b +
[omega]^2 c + [omega]^3 d + [omega]^4 e), [omega] an imaginary fifth
root of unity, or rather the fifth power thereof (a + [omega]b +
[omega]^2 c + [omega]^3d + [omega]^4 e)^5; this is a 24-valued function,
but if we consider the four values corresponding to the roots of unity
[omega], [omega]^2, [omega]^3, [omega]^4, viz. the values
(a + [omega]b + [omega]^2 c + [omega]^3 d + [omega]^4 e)^5,
(a + [omega]^2 b + [omega]^4 c + [omega]d + [omega]^3e)^5,
(a + [omega]^3 b + [omega]c + [omega]^4 d + [omega]^2e)^5,
(a + [omega]^4 b + [omega]^3 c + [omega]^2 d + [omega]e)^5,
any symmetrical function of these, for instance their sum, is a 6-valued
function of the roots, and may therefore be determined by means of a
sextic equation, the coefficients whereof are rational functions of the
coefficients of the original quintic equation; the conclusion being that
the solution of an equation of the fifth order is made to depend upon
that of an equation of the sixth order. This is, of course, useless for
the solution of the quintic equation, which, as already mentioned, does
not admit of solution by radicals; but the equation of the sixth order,
Lagrange's resolvent sextic, is very important, and is intimately
connected with all the later investigations in the theory.
21. It is to be remarked, in regard to the question of solvability by
radicals, that not only the coefficients are taken to be arbitrary, but
it is assumed that they are represented each by a single letter, or say
rather that they are not so expressed in terms of other arbitrary
quantities as to make a solution possible. If the coefficients are not
all arbitrary, for instance, if some of them are zero, a sextic equation
might be of the form x^6 + bx^4 + cx^2 + d = 0, and so be solvable as a
cubic; or if the coefficients of the sextic are given functions of the
six arbitrary quantities a, b, c, d, e, f, such that the sextic is
really of the form (x^2 + ax + b)(x^4 + cx^3 + dx^2 + ex + f) = 0, then
it breaks up into the equations x^2 + ax + b = 0, x^4 + cx^3 + dx^2 + ex
+ f = 0, and is consequently solvable by radicals; so also if the form
is (x -a)(x - b)(x - c)(x - d)(x - e)(x - f) = 0, then the equation is
solvable by radicals,--in this extreme case rationally. Such cases of
solvability are self-evident; but they are enough to show that the
general theorem of the non-solvability by radicals of an equation of the
fifth or any higher order does not in any wise exclude for such orders
the existence of particular equations solvable by radicals, and there
are, in fact, extensive classes of equations which are thus solvable;
the binomial equations x^n - 1 = 0 present an instance.
22. It has already been shown how the several roots of the equation
x^n - 1 = 0 can be expressed in the form cos 2s[pi]/n + i sin
2s[pi]/n, but the question is now that of the algebraical solution (or
solution by radicals) of this equation. There is always a root = 1; if
[omega] be any other root, then obviously [omega], [omega]^2, ...
[omega]^(n - 1) are all of them roots; x^n - 1 contains the factor x -
1, and it thus appears that [omega], [omega]^2, ... [omega]^(n - 1) are
the n - 1 roots of the equation
x^(n - 1) + x^(n - 2) + ... x + 1 = 0;
we have, of course, [omega]^(n - 1) + [omega]^(n - 2) + ... + [omega]
+ 1 = 0.
It is proper to distinguish the cases n prime and n composite; and in
the latter case there is a distinction according as the prime factors
of n are simple or multiple. By way of illustration, suppose
successively n = 15 and n = 9; in the former case, if [alpha] be an
imaginary root of x^3 - 1 = 0 (or root of x^2 + x + 1 = 0), and [beta]
an imaginary root of x^5 - 1 = 0 (or root of x^4 + x^3 + x^2 + x + 1 =
0), then [omega] may be taken = [alpha][beta]; the successive powers
thereof, [alpha][beta], [alpha]^2 [beta]^2, [beta]^3, [alpha][beta]^4,
[alpha]^2, [beta], [alpha][beta]^2, [alpha]^2[beta]^3, [beta]^4,
[alpha], [alpha]^2 [beta], [beta]^2, [alpha][beta]^3, [alpha]^2
[beta]^4, are the roots of x^14 + x^13 + ... + x + 1 = 0; the solution
thus depends on the solution of the equations x^3 - 1 = 0 and x^5 - 1
= 0. In the latter case, if [alpha] be an imaginary root of x^3 - 1 =
0 (or root of x^2 + x + 1 = 0), then the equation x^9 - 1 = 0 gives
x^3 = 1, [alpha], or [alpha]^2; x^3 = 1 gives x = 1, [alpha], or
[alpha]^2; and the solution thus depends on the solution of the
equations x^3 - 1 = 0, x^3 - [alpha] = 0, x^3 - [alpha]^2 = 0. The
first equation has the roots 1, [alpha], [alpha]^2; if [beta] be a
root of either of the others, say if [beta]^3 = [alpha], then assuming
[omega] = [beta], the successive powers are [beta], [beta]^2, [alpha],
[alpha][beta], [alpha][beta]^2, [alpha]^2, [alpha]^2[beta], [alpha]^2
[beta]^2, which are the roots of the equation x^8 + x^7 + ... + x + 1
= 0.
It thus appears that the only case which need be considered is that of
n a prime number, and writing (as is more usual) r in place of
[omega], we have r, r^2, r^3, ... r^(n - 1) as the (n - 1) roots of
the reduced equation
x^(n - 1) + x^(n - 2) + ... + x + 1 = 0;
then not only r^n - 1 = 0, but also r^(n - 1) + r^(n - 2) + ... + r +
1 = 0.
23. The process of solution due to Karl Friedrich Gauss (1801) depends
essentially on the arrangement of the roots in a certain order, viz. not
as above, with the indices of r in arithmetical progression, but with
their indices in geometrical progression; the prime number n has a
certain number of prime roots g, which are such that g^(n - 1) is the
lowest power of g, which is [equivalent to] 1 to the modulus n; or, what
is the same thing, that the series of powers 1, g, g^2, ... g^(n - 2),
each divided by n, leave (in a different order) the remainders 1, 2, 3,
... n - 1; hence giving to r in succession the indices 1, g, g^2, ...
g^(n - 2), we have, in a different order, the whole series of roots r,
r^2, r^3, ... r^(n - 1).
In the most simple case, n = 5, the equation to be solved is x^4 + x^3
+ x^2 + x + 1 = 0; here 2 is a prime root of 5, and the order of the
roots is r, r^2, r^4, r^3. The Gaussian process consists in forming an
equation for determining the periods P1, P2, = r + r^4 and r^2 + r^3
respectively;--these being such that the symmetrical functions P1 +
P2, P1P2 are rationally determinable: in fact P1 + P2 = -1, P1P2 = (r
+ r^4)(r^2 + r^3), = r^3 + r^4 + r^6 + r^7, = r^3 + r^4 + r + r^2, =
-1. P1, P2 are thus the roots of u^2 + u - 1 = 0; and taking them to
be known, they are themselves broken up into subperiods, in the
present case single terms, r and r^4 for P1, r^2 and r^3 for P2; the
symmetrical functions of these are then rationally determined in terms
of P1 and P2; thus r + r^4 = P1, r.r^4 = 1, or r, r^4 are the roots of
u^2 - P1u + 1 = 0. The mode of division is more clearly seen for a
larger value of n; thus, for n = 7 a prime root is = 3, and the
arrangement of the roots is r, r^3, r^2, r^6, r^4, r^5. We may form
either 3 periods each of 2 terms, P1, P2, P3 = r + r^6, r^3 + r^4, r^2
+ r^5 respectively; or else 2 periods each of 3 terms, P1, P2 = r +
r^2 + r^4, r^3 + r^6 + r^5 respectively; in each ease the symmetrical
functions of the periods are rationally determinable: thus in the case
of the two periods P1 + P2 = -1, P1P2 = 3 + r + r^2 + r^3 + r^4 + r^5
+ r^6, = 2; and the periods being known the symmetrical functions of
the several terms of each period are rationally determined in terms of
the periods, thus r + r^2 + r^4 = P1, r.r^2 + r.r^4 + r^2.r^4 = P2,
r.r^2.r^4 = 1.
The theory was further developed by Lagrange (1808), who, applying his
general process to the equation in question, x^(n - 1) + x^(n - 2) + ...
+ x + 1 = 0 (the roots a, b, c ... being the several powers of r, the
indices in geometrical progression as above), showed that the function
(a + [omega]b + [omega]^2 c + ...)^(n - 1) was in this case a given
function of [omega] with integer coefficients.
Reverting to the before-mentioned particular equation x^4 + x^3 + x^2
+ x + 1 = 0, it is very interesting to compare the process of solution
with that for the solution of the general quartic the roots whereof
are a, b, c, d.
Take [omega], a root of the equation [omega]^4 - 1 = 0 (whence [omega]
is = 1, -1, i, or -i, at pleasure), and consider the expression
(a + [omega]b + [omega]^2 c + [omega]^3 d)^4,
the developed value of this is
= a^4 + b^4 + c^4 + d^4 + 6(a^2 c^2 + b^2 d^2) + 12(a^2 bd + b^2 ca + c^2 db + d^2ac)
+[omega] {4(a^3 b + b^3 c + c^3 + d^3 a) + 12(a^2 cd + b^2 da + c^2 ab + d^2 bc)}
+[omega]^2{6(a^2 b^2 + b^2 c^2 + c^2 d^2 + d^2 a^2) + 4(a^3 c + b^3 d + c^3 a + d^3 b) + 24abcd}
+[omega]^3{4(a^3 d + b^3 a + c^3 b + d^3 c) + 12(a^2 bc + b^2 cd + c^2 da + d^2 ab)}
that is, this is a 6-valued function of a, b, c, d, the root of a
sextic (which is, in fact, solvable by radicals; but this is not here
material).
If, however, a, b, c, d denote the roots r, r^2, r^4, r^3 of the
special equation, then the expression becomes
r^4 + r^3 + r + r^2 + 6(1 + 1)+12(r^2 + r^4 + r^3 + r)
+ [omega] {4(1 + 1 + 1 + 1) + 12(r^4 + r^3 + r + r^2)}
+ [omega]^2{6(r + r^2 + r^4 + r^3) + 4(r^2 + r^4 + r^3 + r)}
+ [omega]^3{4(r + r^2 + r^4 + r^3) + 12(r^3 + r + r^2 + r^4)}
viz. this is
= -1 + 4[omega] + 14[omega]^2 - 16[omega]^3,
a completely determined value. That is, we have
(r + [omega]r^2 + [omega]^2 r^4 + [omega]^3 r^3) = -1 + 4[omega] +
14[omega]^2 - 16[omega]^3,
which result contains the solution of the equation. If [omega] = 1, we
have (r + r^2 + r^4 + r^3)^4 = 1, which is right; if [omega] = -1,
then (r + r^4 - r^2 - r^3)^4 = 25; if [omega] = i, then we have {r -
r^4 + i(r^2 - r^3)}^4 = -15 + 20i; and if [omega] = -i, then {r - r^4
- i(r^2 - r^3)}^4 = -15 - 20i; the solution may be completed without
difficulty.
The result is perfectly general, thus:--n being a prime number, r a root
of the equation x^(n - 1) + x^(n - 2) + ... + x + 1 = 0, [omega] a root
of [omega]^(n - 1) - 1 = 0, and g a prime root of g^(n - 1) [equivalent]
1 (mod. n), then
{r + [omega]r^g + ... + [omega]^(n - 2) r^g^(n - 2)}^(n - 1)
is a given function M0 + M1[omega] ... + M_(n - 2)[omega]^(n - 2) with
integer coefficients, and by the extraction of (n - 1)th roots of this
and similar expressions we ultimately obtain r in terms of [omega],
which is taken to be known; the equation x^n - 1 = 0, n a prime number,
is thus solvable by radicals. In particular, if n - 1 be a power of 2,
the solution (by either process) requires the extraction of square roots
only; and it was thus that Gauss discovered that it was possible to
construct geometrically the regular polygons of 17 sides and 257 sides
respectively. Some interesting developments in regard to the theory were
obtained by C.G.J. Jacobi (1837); see the memoir "Ueber die
Kreistheilung, u.s.w.," _Crelle_, t. xxx. (1846).
The equation x^(n - 1) + ... + x + 1 = 0 has been considered for its own
sake, but it also serves as a specimen of a class of equations solvable
by radicals, considered by N.H. Abel (1828), and since called Abelian
equations, viz. for the Abelian equation of the order n, if x be any
root, the roots are x, [theta]x, [theta]^2 x, ... [theta]^(n - 1)x
([theta]x being a rational function of x, and [theta]^nx = x); the
theory is, in fact, very analogous to that of the above particular case.
A more general theorem obtained by Abel is as follows:--If the roots
of an equation of any order are connected together in such wise that
_all_ the roots can be expressed rationally in terms of any one of
them, say x; if, moreover, [theta]x, [theta]1x being any two of the
roots, we have [theta][theta]1x = [theta]1[theta]x, the equation will
be solvable algebraically. It is proper to refer also to Abel's
definition of an _irreducible_ equation:--an equation [phi]x = 0, the
coefficients of which are rational functions of a certain number of
known quantities a, b, c ..., is called irreducible when it is
impossible to express its roots by an equation of an inferior degree,
the coefficients of which are also rational functions of a, b, c ...
(or, what is the same thing, when [phi]x does not break up into
factors which are rational functions of a, b, c ...). Abel applied his
theory to the equations which present themselves in the division of
the elliptic functions, but not to the modular equations.
24. But the theory of the algebraical solution of equations in its most
complete form was established by Evariste Galois (born October 1811,
killed in a duel May 1832; see his collected works, _Liouville_, t. xl.,
1846). The definition of an irreducible equation resembles Abel's,--an
equation is reducible when it admits of a rational divisor, irreducible
in the contrary case; only the word _rational_ is used in this extended
sense that, in connexion with the coefficients of the given equation, or
with the irrational quantities (if any) whereof these are composed, he
considers any number of other irrational quantities called "adjoint
radicals," and he terms rational any rational function of the
coefficients (or the irrationals whereof they are composed) and of these
adjoint radicals; the epithet irreducible is thus taken either
absolutely or in a relative sense, according to the system of adjoint
radicals which are taken into account. For instance, the equation x^4 +
x^3 + x^2 + x + 1 = 0; the left hand side has here no rational divisor,
and the equation is irreducible; but this function is = (x^2 + 1/2 x +
1)^2 -(5/4)x^2, and it has thus the irrational divisors x^2 + 1/2(1 +
[root]5)x + 1, x^2 + 1/2(1 - [root]5)x + 1; and these, if we _adjoin_
the radical [root]5, are rational, and the equation is no longer
irreducible. In the case of a given equation, assumed to be irreducible,
the problem to solve the equation is, in fact, that of finding radicals
by the adjunction of which the equation becomes reducible; for instance,
the general quadric equation x^2 + px + q = 0 is irreducible, but it
becomes reducible, breaking up into rational linear factors, when we
adjoin the radical [root](1/4 p^2 - q).
The fundamental theorem is the Proposition I. of the "Memoire sur les
conditions de resolubilite des equations par radicaux"; viz. given an
equation of which a, b, c ... are the m roots, there is always a group
of permutations of the letters a, b, c ... possessed of the following
properties:--
1. Every function of the roots invariable by the substitutions of the
group is rationally known.
2. Reciprocally every rationally determinable function of the roots is
invariable by the substitutions of the group.
Here by an invariable function is meant not only a function of which
the form is invariable by the substitutions of the group, but further,
one of which the value is invariable by these substitutions: for
instance, if the equation be [phi](x) = 0, then [phi](x) is a function
of the roots invariable by any substitution whatever. And in saying
that a function is rationally known, it is meant that its value is
expressible rationally in terms of the coefficients and of the adjoint
quantities.
For instance in the case of a general equation, the group is simply
the system of the 1.2.3 ... n permutations of all the roots, since, in
this case, the only rationally determinable functions are the
symmetric functions of the roots.
In the case of the equation x^(n - 1) ... + x + 1 = 0, n a prime
number, a, b, c ... k = r, r^g, r^g^2 ... r^g^(n - 2), where g is a
prime root of n, then the group is the cyclical group abc ... k, bc
... ka, ... kab ... j, that is, in this particular case the number of
the permutations of the group is equal to the order of the equation.
This notion of the group of the original equation, or of the group of
the equation as varied by the adjunction of a series of radicals,
seems to be the fundamental one in Galois's theory. But the problem of
solution by radicals, instead of being the sole object of the theory,
appears as the first link of a long chain of questions relating to the
transformation and classification of irrationals.
Returning to the question of solution by radicals, it will be readily
understood that by the adjunction of a radical the group may be
diminished; for instance, in the case of the general cubic, where the
group is that of the six permutations, by the adjunction of the square
root which enters into the solution, the group is reduced to abc, bca,
cab; that is, it becomes possible to express rationally, in terms of
the coefficients and of the adjoint square root, any function such as
a^2 b + b^2 c + c^2 a which is not altered by the cyclical
substitution a into b, b into c, c into a. And hence, to determine
whether an equation of a given form is solvable by radicals, the
course of investigation is to inquire whether, by the successive
adjunction of radicals, it is possible to reduce the original group of
the equation so as to make it ultimately consist of a single
permutation.
The condition in order that an equation of a given prime order n may
be solvable by radicals was in this way obtained--in the first
instance in the form (scarcely intelligible without further
explanation) that every function of the roots x1, x2 ... x_n,
invariable by the substitutions x_(ak + b) for x_k, must be rationally
known; and then in the equivalent form that the resolvent equation of
the order 1.2 ... (n - 2) must have a rational root. In particular,
the condition in order that a quintic equation may be solvable is that
Lagrange's resolvent of the order 6 may have a rational factor, a
result obtained from a direct investigation in a valuable memoir by E.
Luther, _Crelle_, t. xxxiv. (1847).
Among other results demonstrated or announced by Galois may be
mentioned those relating to the modular equations in the theory of
elliptic functions; for the transformations of the orders 5, 7, 11,
the modular equations of the orders 6, 8, 12 are depressible to the
orders 5, 7, 11 respectively; but for the transformation, n a prime
number greater than 11, the depression is impossible.
The general theory of Galois in regard to the solution of equations
was completed, and some of the demonstrations supplied by E. Betti
(1852). See also J.A. Serret's _Cours d'algebre superieure_, 2nd ed.
(1854); 4th ed. (1877-1878).
25. Returning to quintic equations, George Birch Jerrard (1835)
established the theorem that the general quintic equation is by the
extraction of only square and cubic roots reducible to the form x^5 + ax
+ b = 0, or what is the same thing, to x^5 + x + b = 0. The actual
reduction by means of Tschirnhausen's theorem was effected by Charles
Hermite in connexion with his elliptic-function solution of the quintic
equation (1858) in a very elegant manner. It was shown by Sir James
Cockle and Robert Harley (1858-1859) in connexion with the Jerrardian
form, and by Arthur Cayley (1861), that Lagrange's resolvent equation of
the sixth order can be replaced by a more simple sextic equation
occupying a like place in the theory.
The theory of the modular equations, more particularly for the case n =
5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the
case n = 5, the modular equation of the order 6 depends, as already
mentioned, on an equation of the order 5; and conversely the general
quintic equation may be made to depend upon this modular equation of the
order 6; that is, assuming the solution of this modular equation, we can
solve (not by radicals) the general quintic equation; this is Hermite's
solution of the general quintic equation by elliptic functions (1858);
it is analogous to the before-mentioned trigonometrical solution of the
cubic equation. The theory is reproduced and developed in Brioschi's
memoir, "Uber die Auflosung der Gleichungen vom funften Grade," _Math.
Annalen_, t. xiii. (1877-1878).
26. The modern work, reproducing the theories of Galois, and
exhibiting the theory of algebraic equations as a whole, is C.
Jordan's _Traite des substitutions et des equations algebriques_
(Paris, 1870). The work is divided into four books--book i.,
preliminary, relating to the theory of congruences; book ii. is in two
chapters, the first relating to substitutions in general, the second
to substitutions defined analytically, and chiefly to linear
substitutions; book iii. has four chapters, the first discussing the
principles of the general theory, the other three containing
applications to algebra, geometry, and the theory of transcendents;
lastly, book iv., divided into seven chapters, contains a
determination of the general types of equations solvable by radicals,
and a complete system of classification of these types. A glance
through the index will show the vast extent which the theory has
assumed, and the form of general conclusions arrived at; thus, in book
iii., the algebraical applications comprise Abelian equations,
equations of Galois; the geometrical ones comprise Q. Hesse's
equation, R.F.A. Clebsch's equations, lines on a quartic surface
having a nodal line, singular points of E.E. Kummer's surface, lines
on a cubic surface, problems of contact; the applications to the
theory of transcendents comprise circular functions, elliptic
functions (including division and the modular equation), hyperelliptic
functions, solution of equations by transcendents. And on this last
subject, solution of equations by transcendents, we may quote the
result--"the solution of the general equation of an order superior to
five cannot be made to depend upon that of the equations for the
division of the circular or elliptic functions"; and again (but with a
reference to a possible case of exception), "the general equation
cannot be solved by aid of the equations which give the division of
the hyperelliptic functions into an odd number of parts." (See also
GROUPS, THEORY OF.) (A. Ca.)
BIBLIOGRAPHY.--For the general theory see W.S. Burnside and A.W.
Panton, _The Theory of Equations_ (4th ed., 1899-1901); the Galoisian
theory is treated in G.B. Matthews, _Algebraic Equations_ (1907). See
also the _Ency. d. math. Wiss._ vol. ii.