Page:EB1911 - Volume 19.djvu/890

product of the contents of two forms is equal to the content of the product of the forms. Every form is associated with a definite ideal 𝔪, and we have 𝖭(𝖥)＝𝖭(𝔪) if 𝔪 is the content of 𝖥, and 𝖭(𝔪) has the meaning already assigned to it. On the other hand, to a given ideal correspond an indefinite number of forms of which it is the content; for instance (§ 46, end) we can find forms 𝛼𝑥＋𝛽𝑦 of which any given ideal is the content.

58. Now let 𝜔₁, 𝜔₂, . . . 𝜔_𝒏, be a basis of 𝔬; 𝑢₁, 𝑢₂, . . . 𝑢_𝒏 a set of indeterminates; and

𝜉＝𝜔₁𝑢₁＋𝜔₂𝑢₂＋ . . . ＋𝜔_𝒏𝑢_𝒏:

𝜉 is called the fundamental form of Ω. It satisfies the equation 𝖭(𝑥－𝜉)＝0, or

𝖥(𝑥)＝𝑥^𝒏＋𝖴₁𝑥^𝒏－1＋ . . . ＋𝖴_𝒏＝0

where 𝖴₁, 𝖴₂, . . . 𝖴_𝒏 are rational polynomials in 𝑢₁, 𝑢₂, . . .𝑢_𝒏 with rational integral coefficients. This is is called the fundamental equation.

Suppose now that 𝑝 is a rational prime, and that 𝑝＝𝔭^𝒂𝔮^𝒃𝔯^𝒄 where 𝔭, 𝔮, 𝔯, . . . &c., are the different ideal prime factors of 𝑝, then if 𝖥(𝑥) is the left-hand side of the fundamental equation there is an identical congruence

𝖥(𝑥)＝{𝖯(𝑥)}^𝒂{𝖰(𝑥)}^𝒃{𝖱(𝑥)}^𝒄 . . . (mod 𝑝)

where 𝖯(𝑥), 𝖰(𝑥), &c., are prime functions with respect to 𝔭. The meaning of this is that if we expand the expression on the right-hand side of the congruence, the coefficient of every term 𝑥^𝑙𝑢₁^𝑚. . . 𝑢_𝒏^𝑡 will be congruent, mod 𝑝, to the corresponding coefficient in 𝖥(𝑥). If 𝑓, 𝑔, ℎ, &c., are the degrees of 𝔭, 𝔮, 𝔯, &c. (§ 47), then 𝑓, 𝑔, ℎ, . . . are the dimensions in 𝑥, 𝑢₁, 𝑢₂,. . . 𝑢_𝒏 of the forms of 𝖯, 𝖰, 𝖱, respectively. For every prime 𝑝, which is not a factor of 𝚫, 𝒂＝𝒃＝𝒄＝. . .＝1 and 𝖥(𝑥) is congruent to the product of a set of different prime factors, as many in number as there are different ideal prime factors of 𝑝. In particular, if 𝑝 is a prime in Ω, 𝖥(𝑥) is a prime function (mod 𝑝) and conversely.

It generally happens that rational integral values 𝒂₁, 𝒂₂, . . . 𝒂_𝒏 can be assigned to 𝑢₁, 𝑢₂, . . . 𝑢_𝒏 such that 𝖴_𝒏, the last term in the fundamental equation, then has a value which is prime to 𝑝. Supposing that this condition is satisfied, let 𝒂₁𝜔₁＋𝒂₂𝜔₂＋ . . .＋𝒂_𝒏𝜔_𝒏＝𝛼; and let 𝖯₁(𝛼) be the result of putting 𝑥＝𝛼, 𝑢_𝑖＝𝒂_𝑖 in 𝖯(𝑥). Then the ideal 𝔭 is completely determined as the greatest common divisor of 𝑝 and 𝖯₁(𝛼); and similarly for the other prime factors of 𝑝. There are, however, exceptional cases when the condition above stated is not satisfied.

59. Cyclotomy.—It follows from de Moivre’s theorem that the arithmetical solution of the equation 𝑥^𝑚－1＝0 corresponds to the division of the circumference of a circle into 𝑚 equal parts. The case when 𝑚 is composite is easily made to depend on that where 𝑚 is a power of a prime; if 𝑚 is a power of 2, the solution is effected by a chain of quadratic equations, and it only remains to consider the case when 𝑚＝𝑞^𝜅, a power of an odd prime. It will be convenient to write 𝜇＝𝜙(𝑚)＝𝑞^𝜅－1(𝑞－1); if we also put 𝑟＝𝑒^{2𝜋𝑖∕𝑚}, the primitive roots of 𝑥^𝑚＝1 will be 𝜇 in number, and represented by 𝑟, 𝑟^𝒂, 𝑟^𝒃, &c. where 1, 𝒂, 𝒃, &c., form a complete set of prime residues to the modulus 𝑚. These will be the roots of an irreducible equation 𝑓(𝑥)＝0 of degree 𝜇; the symbol 𝑓(𝑥) denoting (𝑥^𝑚－1)÷(𝑥^𝑚/𝑞－1). There are primitive roots of the congruence 𝑥^𝜇＝1 (mod 𝑚); let 𝑔 be any one of these. Then if we put 𝑟^𝑔ℎ＝𝑟_ℎ, we obtain all the roots of 𝑓(𝑥)＝0 in a definite cyclical order (𝑟₁, 𝑟₂ . . .𝑟_𝜇; and the change of 𝑟 into 𝑟^𝑔 produces a cyclical permutation of the roots. It follows from this that every cyclic polynomial in 𝑟₁, 𝑟₂ . . .𝑟_𝜇 with rational coefficients is equal to a rational number. Thus if we write 𝑙＋𝒂𝑔＋𝒃𝑔²＋.＋𝑘𝑔^𝜇－1＝𝒏, we have, in virtue of 𝑟_ℎ＝𝑟^𝑔𝑘, 𝑟₁^𝒂𝑟₂^𝒃𝑟_𝜇－1^𝑘𝑟_𝜇^𝑙＝𝑟^𝒏, and, if we use 𝖲 to denote cyclical summation, 𝖲(𝑟₁^𝒂𝑟₂^𝒃. . .𝑟_𝜇^𝑙)＝𝑟^𝒏＋𝑟^𝒏𝑔＋ . . . ＋𝑟^{𝒏𝑔𝜇－1}, the sum of the 𝒏th powers of all the roots of 𝑓(𝑥)＝0, and this is a rational integer or zero. Since every cyclic polynomial is the sum of parts similar to 𝖲(𝑟₁^𝒂𝑟₂^𝒃. . .𝑟_𝜇^𝑙), the theorem is proved. Now let 𝑒, 𝑓 be any two conjugate factors of 𝜇, so that 𝑒𝑓＝𝜇, and let

𝜂_𝑖＝𝑟_𝑖＋𝑟_𝑖＋𝑒＋𝑟_𝑖＋2𝑒＋. . . ＋𝑟_{𝑖＋(𝑓－1)𝑒}       (𝑖＝1, 2,. . .𝑒)

then the elementary symmetric functions 𝚺𝜂_𝑖, 𝚺𝜂_𝑖𝜂^𝑗, &c., are cyclical functions of the roots of 𝑓(𝑥)＝0 and therefore have rational values which can be calculated: consequently 𝜂₁, 𝜂₂, . . .𝜂_𝑒, which are called the 𝑓-nomial periods, are the roots of an equation 𝖥(𝜂)＝𝜂^𝑒＋𝒄₁𝜂^𝑒－1＋ . . . ＋𝒄_𝑒＝0 with rational integral coefficients. This is irreducible, and defines a field of order 𝑒 contained in the field defined by 𝑓(𝑥)＝0. Moreover, the change of 𝑟 into 𝑟^𝑔 alters 𝒏_𝑖 into 𝜂_𝑖＋1, and we have the theorem that any cyclical function of 𝜂₁, 𝜂₂, . . . 𝒏_𝑒 is rational. Now let ℎ, 𝑘 be any conjugate factors of 𝑓 and put

𝑧_𝑖＝𝑟_𝑖＋𝑟_{𝑖＋ℎ𝑒}＋𝑟^𝑖_＋2ℎ𝑒＋ . . . 𝑟^𝑖_{＋(𝑓－ℎ)𝑒}      (𝑖＝1, 2, 3,)

then 𝜁₁,𝜁_1＋𝑒,𝜁_1＋2𝑒. . .𝜁_{1＋(ℎ－1)𝑒} will be the roots of an equation

𝖦(𝜁)＝𝜁^ℎ－𝜂₁𝜁^ℎ－1＋𝒄₂𝜁^ℎ－2＋ . . . ＋𝒄_ℎ＝0

the coefficients of which are expressible as rational polynomials in 𝜂₁. Dividing ℎ into two conjugate factors, we can deduce from 𝖦(𝜁)＝0 another period equation, the coefficients of which are rational polynomials in 𝜂₂, 𝜁₁, and so on. By choosing for 𝑒, ℎ, &c., the successive prime factors of 𝜇, ending up with 2, we obtain a set of equations of prime degree, each rational in the roots of the preceding equations, and the last having 𝑟₁ and 𝑟₁^－1 for its roots. Thus to take a very interesting historical case, let 𝑚＝17, so that 𝜇＝16＝2⁴, the equations are all quadratics, and if we take 3 as the primitive root of 17, they are

𝜂²＋𝜂－4＝0,        𝜁²－𝜂𝜁－1＝0

2𝜆²－2𝜁𝜆＋(𝜂𝜁－𝜂＋𝜁－3)＝0,   𝜌²－𝜆𝜌＋1＝0.

If two quantities (real or complex) 𝒂 and 𝒃 are represented in the usual way by points in a plane, the roots of 𝑥²＋𝒂𝑥＋𝒃＝0 will be represented by two points which can be found by a Euclidean construction, that is to say, one requiring only the use of rule and compass. Hence a regular polygon of seventeen sides can be inscribed in a given circle by means of a Euclidean construction; a fact first discovered by Gauss, who also found the general law, which is that a regular polygon of 𝑚 sides can be inscribed in a circle by Euclidean construction if and only if 𝜙(𝑚) is a power of 2; in other words 𝑚＝2^𝜅𝖯 where 𝖯 is a product of different odd primes, each of which is of the form 2^𝒏＋1.

Returning to the case 𝑚＝𝑞^𝜅, we shall call the chain of equations 𝖥(𝜂)＝0, &c., when each is of prime degree, a set of Galoisian auxiliaries. We can find different sets, because in forming them we can take the prime factors of 𝜇 in any order we like; but their number is always the same, and their degrees always form the same aggregate, namely, the prime factors of 𝜇. No other chain of auxiliaries having similar properties can be formed containing fewer equations of a given prime degree 𝑝; a fact first stated by Gauss, to whom this theory is mainly due. Thus if 𝑚＝𝑞^𝜅 we must have at least (𝜅－1) auxiliaries of order 𝑞, and if 𝑞－1＝2^𝛼𝑝^𝛽 . . ., we must also have 𝛼 quadratics, 𝛽 equations of order 𝑝, an so on. For this reason a set of Galoisian auxiliaries may be regarded as providing the simplest solution of the equation 𝑓(𝑥)＝0.

60. When 𝑚 is an odd prime 𝑝, there is another very interesting way of solving the equation (𝑥^𝑝－1)÷(𝑥－1)＝0. As before let (𝑟₁, 𝑟₂, . . . 𝑟_𝑝－1) be its roots arranged in a cycle by means of a primitive root of 𝑥^𝑝－1≡1 (mod 𝑝); and let 𝜖 be a primitive root of 𝜖^𝑝－1＝1. Also let

𝜃₁＝𝑟₁＋𝜖𝑟₂＋𝜖²𝑟₃＋ . . . ＋𝜖𝑝－2𝑟𝑝－1
𝜃𝑘＝𝑟₁＋𝜖𝑘𝑟₂＋𝜖2𝑘𝑟₃＋ . . . ＋𝜖－𝑘𝑟𝑝－1	(𝑘＝2, 3, . . . 𝑝－2)

so that 𝜃_𝑘 is derived from 𝜃₁ by changing 𝜖 into 𝜖^𝑘.

The cyclical permutation (𝑟₁, 𝑟₂, . . .𝑟_𝑝－1) applied to 𝜃_𝑘 converts it into 𝜖^－𝑘𝜃_𝑘; hence 𝜃₁𝜃_𝑘/𝜃_𝑘＋1 is unaltered, and may be expressed as a rational, and therefore as an integral function of 𝜖. It is found by calculation that we may put

$\psi _{k}(\epsilon )={\theta _{1}\theta _{k} \over \theta _{k+1}}=\sum _{m=2}^{m=p+1}\epsilon ^{{\mbox{ind}}\ m+k\,{\mbox{ind}}(p+1-m)}\qquad [k=1,\ 2,\,.\,.\,.\,(p-3)]$

while

𝜃₁𝜃_𝑝－2＝－𝑝.

In the exponents of 𝜓_𝑘(𝜖) the indices are taken to the base 𝑔 used to establish the cyclical order (𝑟₁, 𝑟₂, . . . 𝑟_𝑝－1). Multiplying together the (𝑝－2) preceding equalities, the result is

𝜃₁^𝑝－1＝－𝑝𝜓₁(𝜖)𝜓₂(𝜖) . . . 𝜓_𝑝－3(𝜖)＝𝖱(𝜖)

where 𝖱(𝜖) is a rational integral function of 𝜖 the degree of which, in its reduced form, is less than 𝜙(𝑝－1). Let 𝜌 be any one definite root of 𝑥^𝑝－1＝𝖱(𝜖), and put 𝜃₁＝𝜌: then since

${\theta _{1}^{k} \over \theta _{k}}=\psi _{1}\psi _{2}...\psi _{k-1}$

we must take 𝜃_𝑘＝𝜌^𝑘/𝜓₁𝜓₂ . . . 𝜓_𝑘－1＝𝖱_𝑘(𝜖)𝜌^𝑘 , where 𝖱_𝑘(𝜖) is a rational function of 𝜖, which we may suppose put into its reduced integral form; and finally, by addition of the equations which define 𝜃₁, 𝜃₂, &c.,

(𝑝－1)𝑟₁＝𝜌＋𝖱₂(𝜖)𝜌²＋𝖱₃(𝜖)𝜌³＋ . . . ＋𝖱_𝑝－2(𝜖)𝜌^𝑝－2.

If in this formula we change 𝜌 into 𝜖^－ℎ𝜌, and 𝑟₁ into 𝑟_ℎ＋1, it still remains true.

It will be observed that this second mode of solution employs a Lagrangian resolvent 𝜃₁; considered merely as a solution it is neither so direct nor so fundamental as that of Gauss. But the form of the solution is very interesting; and the auxiliary numbers 𝜓(𝜖) have many curious properties, which have been investigated by Jacobi, Cauchy and Kronecker.

61. When 𝑚＝𝑞^𝜅, the discriminant of the corresponding cyclotomic field is ±𝑞^𝜆, where 𝜆＝𝑞^𝜅－1(𝜅𝑞－𝜅－1). The prime 𝑞 is equal to 𝔮^𝜇, where 𝜇＝𝜙(𝑚)＝𝑞^𝜅－1(𝑞－1), and 𝔮 is a prime ideal of the first degree. If 𝑝 is any rational prime distinct from 𝑔, and 𝑓 the least exponent such that 𝑝^𝑓≡1 (mod. 𝑚), 𝑓 will be a factor of 𝜇, and putting 𝜇/𝑓＝𝑒, we have 𝑝＝𝔭₁𝔭₂ . . .𝔭_𝑒, where 𝔭₁, 𝔭₂ . . . 𝔭_𝑒 are different prime ideals each of the 𝑓th degree. There are similar theorems for the case when 𝑚 is divisible by more than one rational prime.

Kummer has stated and proved laws of reciprocity for quadratic and higher residues in what are called regular fields, the definition of which is as follows. Let the field be 𝖱(𝑒^{2𝜋𝑖/𝑝}), where 𝑝 is an odd prime; then this field is regular, and 𝑝 is said to be a regular prime,

when ℎ, the number of ideal classes in the field, is not divisible by 𝑝. Kummer proved the very curious fact that 𝑝 is regular if, and only if, it is not a factor of the denominators of the first 1/2(𝑝－3) Bernoullian