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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 ๐”ช, and we have ๐–ญ(๐–ฅ)๏ผ๐–ญ(๐”ช) if ๐”ช is the content of ๐–ฅ, and ๐–ญ(๐”ช) 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 (ยง 46, end) we can find forms ๐›ผ๐‘ฅ๏ผ‹๐›ฝ๐‘ฆ of which any given ideal is the content.

58. Now let ๐œ”โ‚, ๐œ”โ‚‚, . . . ๐œ”๐’, be a basis of ๐”ฌ; ๐‘ขโ‚, ๐‘ขโ‚‚, . . . ๐‘ข๐’ a set of indeterminates; and

๐œ‰๏ผ๐œ”โ‚๐‘ขโ‚๏ผ‹๐œ”โ‚‚๐‘ขโ‚‚๏ผ‹ . . . ๏ผ‹๐œ”๐’๐‘ข๐’:


๐œ‰ is called the fundamental form of ฮฉ. It satisfies the equation ๐–ญ(๐‘ฅ๏ผ๐œ‰)๏ผ0, or

๐–ฅ(๐‘ฅ)๏ผ๐‘ฅ๐’๏ผ‹๐–ดโ‚๐‘ฅ๐’๏ผ1๏ผ‹ . . . ๏ผ‹๐–ด๐’๏ผ0


where ๐–ดโ‚, ๐–ดโ‚‚, . . . ๐–ด๐’ are rational polynomials in ๐‘ขโ‚, ๐‘ขโ‚‚, . . .๐‘ข๐’ with rational integral coefficients. This is is called the fundamental equation.

Suppose now that ๐‘ is a rational prime, and that ๐‘๏ผ๐”ญ๐’‚๐”ฎ๐’ƒ๐”ฏ๐’„ where ๐”ญ, ๐”ฎ, ๐”ฏ, . . . &c., are the different ideal prime factors of ๐‘, then if ๐–ฅ(๐‘ฅ) is the left-hand side of the fundamental equation there is an identical congruence

๐–ฅ(๐‘ฅ)๏ผ{๐–ฏ(๐‘ฅ)}๐’‚{๐–ฐ(๐‘ฅ)}๐’ƒ{๐–ฑ(๐‘ฅ)}๐’„ . . . (mod ๐‘)


where ๐–ฏ(๐‘ฅ), ๐–ฐ(๐‘ฅ), &c., are prime functions with respect to ๐”ญ. The meaning of this is that if we expand the expression on the right-hand side of the congruence, the coefficient of every term ๐‘ฅ๐‘™๐‘ขโ‚๐‘š. . . ๐‘ข๐’๐‘ก will be congruent, mod ๐‘, to the corresponding coefficient in ๐–ฅ(๐‘ฅ). If ๐‘“, ๐‘”, โ„Ž, &c., are the degrees of ๐”ญ, ๐”ฎ, ๐”ฏ, &c. (ยง 47), then ๐‘“, ๐‘”, โ„Ž, . . . are the dimensions in ๐‘ฅ, ๐‘ขโ‚, ๐‘ขโ‚‚,. . . ๐‘ข๐’ of the forms of ๐–ฏ, ๐–ฐ, ๐–ฑ, respectively. For every prime ๐‘, which is not a factor of ๐šซ, ๐’‚๏ผ๐’ƒ๏ผ๐’„๏ผ. . .๏ผ1 and ๐–ฅ(๐‘ฅ) is congruent to the product of a set of different prime factors, as many in number as there are different ideal prime factors of ๐‘. In particular, if ๐‘ is a prime in ฮฉ, ๐–ฅ(๐‘ฅ) is a prime function (mod ๐‘) and conversely.

It generally happens that rational integral values ๐’‚โ‚, ๐’‚โ‚‚, . . . ๐’‚๐’ can be assigned to ๐‘ขโ‚, ๐‘ขโ‚‚, . . . ๐‘ข๐’ such that ๐–ด๐’, the last term in the fundamental equation, then has a value which is prime to ๐‘. Supposing that this condition is satisfied, let ๐’‚โ‚๐œ”โ‚๏ผ‹๐’‚โ‚‚๐œ”โ‚‚๏ผ‹ . . .๏ผ‹๐’‚๐’๐œ”๐’๏ผ๐›ผ; and let ๐–ฏโ‚(๐›ผ) be the result of putting ๐‘ฅ๏ผ๐›ผ, ๐‘ข๐‘–๏ผ๐’‚๐‘– in ๐–ฏ(๐‘ฅ). Then the ideal ๐”ญ is completely determined as the greatest common divisor of ๐‘ and ๐–ฏโ‚(๐›ผ); and similarly for the other prime factors of ๐‘. There are, however, exceptional cases when the condition above stated is not satisfied.

59. Cyclotomy.โ€”It follows from de Moivreโ€™s theorem that the arithmetical solution of the equation ๐‘ฅ๐‘š๏ผ1๏ผ0 corresponds to the division of the circumference of a circle into ๐‘š equal parts. The case when ๐‘š is composite is easily made to depend on that where ๐‘š is a power of a prime; if ๐‘š is a power of 2, the solution is effected by a chain of quadratic equations, and it only remains to consider the case when ๐‘š๏ผ๐‘ž๐œ…, a power of an odd prime. It will be convenient to write ๐œ‡๏ผ๐œ™(๐‘š)๏ผ๐‘ž๐œ…๏ผ1(๐‘ž๏ผ1); if we also put ๐‘Ÿ๏ผ๐‘’2๐œ‹๐‘–โˆ•๐‘š, the primitive roots of ๐‘ฅ๐‘š๏ผ1 will be ๐œ‡ in number, and represented by ๐‘Ÿ, ๐‘Ÿ๐’‚, ๐‘Ÿ๐’ƒ, &c. where 1, ๐’‚, ๐’ƒ, &c., form a complete set of prime residues to the modulus ๐‘š. These will be the roots of an irreducible equation ๐‘“(๐‘ฅ)๏ผ0 of degree ๐œ‡; the symbol ๐‘“(๐‘ฅ) denoting (๐‘ฅ๐‘š๏ผ1)รท(๐‘ฅ๐‘š/๐‘ž๏ผ1). There are primitive roots of the congruence ๐‘ฅ๐œ‡๏ผ1 (mod ๐‘š); let ๐‘” be any one of these. Then if we put ๐‘Ÿ๐‘”โ„Ž๏ผ๐‘Ÿโ„Ž, we obtain all the roots of ๐‘“(๐‘ฅ)๏ผ0 in a definite cyclical order (๐‘Ÿโ‚, ๐‘Ÿโ‚‚ . . .๐‘Ÿ๐œ‡; and the change of ๐‘Ÿ into ๐‘Ÿ๐‘” produces a cyclical permutation of the roots. It follows from this that every cyclic polynomial in ๐‘Ÿโ‚, ๐‘Ÿโ‚‚ . . .๐‘Ÿ๐œ‡ with rational coefficients is equal to a rational number. Thus if we write ๐‘™๏ผ‹๐’‚๐‘”๏ผ‹๐’ƒ๐‘”ยฒ๏ผ‹.๏ผ‹๐‘˜๐‘”๐œ‡๏ผ1๏ผ๐’, we have, in virtue of ๐‘Ÿโ„Ž๏ผ๐‘Ÿ๐‘”๐‘˜, ๐‘Ÿโ‚๐’‚๐‘Ÿโ‚‚๐’ƒ๐‘Ÿ๐œ‡๏ผ1๐‘˜๐‘Ÿ๐œ‡๐‘™๏ผ๐‘Ÿ๐’, and, if we use ๐–ฒ to denote cyclical summation, ๐–ฒ(๐‘Ÿโ‚๐’‚๐‘Ÿโ‚‚๐’ƒ. . .๐‘Ÿ๐œ‡๐‘™)๏ผ๐‘Ÿ๐’๏ผ‹๐‘Ÿ๐’๐‘”๏ผ‹ . . . ๏ผ‹๐‘Ÿ๐’๐‘”๐œ‡๏ผ1, the sum of the ๐’th powers of all the roots of ๐‘“(๐‘ฅ)๏ผ0, and this is a rational integer or zero. Since every cyclic polynomial is the sum of parts similar to ๐–ฒ(๐‘Ÿโ‚๐’‚๐‘Ÿโ‚‚๐’ƒ. . .๐‘Ÿ๐œ‡๐‘™), the theorem is proved. Now let ๐‘’, ๐‘“ be any two conjugate factors of ๐œ‡, so that ๐‘’๐‘“๏ผ๐œ‡, and let

๐œ‚๐‘–๏ผ๐‘Ÿ๐‘–๏ผ‹๐‘Ÿ๐‘–๏ผ‹๐‘’๏ผ‹๐‘Ÿ๐‘–๏ผ‹2๐‘’๏ผ‹. . . ๏ผ‹๐‘Ÿ๐‘–๏ผ‹(๐‘“๏ผ1)๐‘’       (๐‘–๏ผ1, 2,. . .๐‘’)


then the elementary symmetric functions ๐šบ๐œ‚๐‘–, ๐šบ๐œ‚๐‘–๐œ‚๐‘—, &c., are cyclical functions of the roots of ๐‘“(๐‘ฅ)๏ผ0 and therefore have rational values which can be calculated: consequently ๐œ‚โ‚, ๐œ‚โ‚‚, . . .๐œ‚๐‘’, which are called the ๐‘“-nomial periods, are the roots of an equation ๐–ฅ(๐œ‚)๏ผ๐œ‚๐‘’๏ผ‹๐’„โ‚๐œ‚๐‘’๏ผ1๏ผ‹ . . . ๏ผ‹๐’„๐‘’๏ผ0 with rational integral coefficients. This is irreducible, and defines a field of order ๐‘’ contained in the field defined by ๐‘“(๐‘ฅ)๏ผ0. Moreover, the change of ๐‘Ÿ into ๐‘Ÿ๐‘” alters ๐’๐‘– into ๐œ‚๐‘–๏ผ‹1, and we have the theorem that any cyclical function of ๐œ‚โ‚, ๐œ‚โ‚‚, . . . ๐’๐‘’ is rational. Now let โ„Ž, ๐‘˜ be any conjugate factors of ๐‘“ and put

๐‘ง๐‘–๏ผ๐‘Ÿ๐‘–๏ผ‹๐‘Ÿ๐‘–๏ผ‹โ„Ž๐‘’๏ผ‹๐‘Ÿ๐‘–๏ผ‹2โ„Ž๐‘’๏ผ‹ . . . ๐‘Ÿ๐‘–๏ผ‹(๐‘“๏ผโ„Ž)๐‘’      (๐‘–๏ผ1, 2, 3,)


then ๐œโ‚,๐œ1๏ผ‹๐‘’,๐œ1๏ผ‹2๐‘’. . .๐œ1๏ผ‹(โ„Ž๏ผ1)๐‘’ will be the roots of an equation

๐–ฆ(๐œ)๏ผ๐œโ„Ž๏ผ๐œ‚โ‚๐œโ„Ž๏ผ1๏ผ‹๐’„โ‚‚๐œโ„Ž๏ผ2๏ผ‹ . . . ๏ผ‹๐’„โ„Ž๏ผ0


the coefficients of which are expressible as rational polynomials in ๐œ‚โ‚. Dividing โ„Ž into two conjugate factors, we can deduce from ๐–ฆ(๐œ)๏ผ0 another period equation, the coefficients of which are rational polynomials in ๐œ‚โ‚‚, ๐œโ‚, and so on. By choosing for ๐‘’, โ„Ž, &c., the successive prime factors of ๐œ‡, 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 ๐‘Ÿโ‚ and ๐‘Ÿโ‚๏ผ1 for its roots. Thus to take a very interesting historical case, let ๐‘š๏ผ17, so that ๐œ‡๏ผ16๏ผ2โด, the equations are all quadratics, and if we take 3 as the primitive root of 17, they are

๐œ‚ยฒ๏ผ‹๐œ‚๏ผ4๏ผ0,        ๐œยฒ๏ผ๐œ‚๐œ๏ผ1๏ผ0

2๐œ†ยฒ๏ผ2๐œ๐œ†๏ผ‹(๐œ‚๐œ๏ผ๐œ‚๏ผ‹๐œ๏ผ3)๏ผ0,   ๐œŒยฒ๏ผ๐œ†๐œŒ๏ผ‹1๏ผ0.


If two quantities (real or complex) ๐’‚ and ๐’ƒ are represented in the usual way by points in a plane, the roots of ๐‘ฅยฒ๏ผ‹๐’‚๐‘ฅ๏ผ‹๐’ƒ๏ผ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 ๐‘š sides can be inscribed in a circle by Euclidean construction if and only if ๐œ™(๐‘š) is a power of 2; in other words ๐‘š๏ผ2๐œ…๐–ฏ where ๐–ฏ is a product of different odd primes, each of which is of the form 2๐’๏ผ‹1.

Returning to the case ๐‘š๏ผ๐‘ž๐œ…, we shall call the chain of equations ๐–ฅ(๐œ‚)๏ผ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 ๐œ‡ 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 ๐œ‡. No other chain of auxiliaries having similar properties can be formed containing fewer equations of a given prime degree ๐‘; a fact first stated by Gauss, to whom this theory is mainly due. Thus if ๐‘š๏ผ๐‘ž๐œ… we must have at least (๐œ…๏ผ1) auxiliaries of order ๐‘ž, and if ๐‘ž๏ผ1๏ผ2๐›ผ๐‘๐›ฝ . . ., we must also have ๐›ผ quadratics, ๐›ฝ equations of order ๐‘, an so on. For this reason a set of Galoisian auxiliaries may be regarded as providing the simplest solution of the equation ๐‘“(๐‘ฅ)๏ผ0.

60. When ๐‘š is an odd prime ๐‘, there is another very interesting way of solving the equation (๐‘ฅ๐‘๏ผ1)รท(๐‘ฅ๏ผ1)๏ผ0. As before let (๐‘Ÿโ‚, ๐‘Ÿโ‚‚, . . . ๐‘Ÿ๐‘๏ผ1) be its roots arranged in a cycle by means of a primitive root of ๐‘ฅ๐‘๏ผ1โ‰ก1 (mod ๐‘); and let ๐œ– be a primitive root of ๐œ–๐‘๏ผ1๏ผ1. Also let

๐œƒโ‚๏ผ๐‘Ÿโ‚๏ผ‹๐œ–๐‘Ÿโ‚‚๏ผ‹๐œ–ยฒ๐‘Ÿโ‚ƒ๏ผ‹ . . . ๏ผ‹๐œ–๐‘๏ผ2๐‘Ÿ๐‘๏ผ1  
๐œƒ๐‘˜๏ผ๐‘Ÿโ‚๏ผ‹๐œ–๐‘˜๐‘Ÿโ‚‚๏ผ‹๐œ–2๐‘˜๐‘Ÿโ‚ƒ๏ผ‹ . . . ๏ผ‹๐œ–๏ผ๐‘˜๐‘Ÿ๐‘๏ผ1 (๐‘˜๏ผ2, 3, . . . ๐‘๏ผ2)


so that ๐œƒ๐‘˜ is derived from ๐œƒโ‚ by changing ๐œ– into ๐œ–๐‘˜.

The cyclical permutation (๐‘Ÿโ‚, ๐‘Ÿโ‚‚, . . .๐‘Ÿ๐‘๏ผ1) applied to ๐œƒ๐‘˜ converts it into ๐œ–๏ผ๐‘˜๐œƒ๐‘˜; hence ๐œƒโ‚๐œƒ๐‘˜/๐œƒ๐‘˜๏ผ‹1 is unaltered, and may be expressed as a rational, and therefore as an integral function of ๐œ–. It is found by calculation that we may put


while

๐œƒโ‚๐œƒ๐‘๏ผ2๏ผ๏ผ๐‘.


In the exponents of ๐œ“๐‘˜(๐œ–) the indices are taken to the base ๐‘” used to establish the cyclical order (๐‘Ÿโ‚, ๐‘Ÿโ‚‚, . . . ๐‘Ÿ๐‘๏ผ1). Multiplying together the (๐‘๏ผ2) preceding equalities, the result is

๐œƒโ‚๐‘๏ผ1๏ผ๏ผ๐‘๐œ“โ‚(๐œ–)๐œ“โ‚‚(๐œ–) . . . ๐œ“๐‘๏ผ3(๐œ–)๏ผ๐–ฑ(๐œ–)


where ๐–ฑ(๐œ–) is a rational integral function of ๐œ– the degree of which, in its reduced form, is less than ๐œ™(๐‘๏ผ1). Let ๐œŒ be any one definite root of ๐‘ฅ๐‘๏ผ1๏ผ๐–ฑ(๐œ–), and put ๐œƒโ‚๏ผ๐œŒ: then since


we must take ๐œƒ๐‘˜๏ผ๐œŒ๐‘˜/๐œ“โ‚๐œ“โ‚‚ . . . ๐œ“๐‘˜๏ผ1๏ผ๐–ฑ๐‘˜(๐œ–)๐œŒ๐‘˜ , where ๐–ฑ๐‘˜(๐œ–) is a rational function of ๐œ–, which we may suppose put into its reduced integral form; and finally, by addition of the equations which define ๐œƒโ‚, ๐œƒโ‚‚, &c.,

(๐‘๏ผ1)๐‘Ÿโ‚๏ผ๐œŒ๏ผ‹๐–ฑโ‚‚(๐œ–)๐œŒยฒ๏ผ‹๐–ฑโ‚ƒ(๐œ–)๐œŒยณ๏ผ‹ . . . ๏ผ‹๐–ฑ๐‘๏ผ2(๐œ–)๐œŒ๐‘๏ผ2.


If in this formula we change ๐œŒ into ๐œ–๏ผโ„Ž๐œŒ, and ๐‘Ÿโ‚ into ๐‘Ÿโ„Ž๏ผ‹1, it still remains true.

It will be observed that this second mode of solution employs a Lagrangian resolvent ๐œƒโ‚; 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 ๐œ“(๐œ–) have many curious properties, which have been investigated by Jacobi, Cauchy and Kronecker.

61. When ๐‘š๏ผ๐‘ž๐œ…, the discriminant of the corresponding cyclotomic field is ยฑ๐‘ž๐œ†, where ๐œ†๏ผ๐‘ž๐œ…๏ผ1(๐œ…๐‘ž๏ผ๐œ…๏ผ1). The prime ๐‘ž is equal to ๐”ฎ๐œ‡, where ๐œ‡๏ผ๐œ™(๐‘š)๏ผ๐‘ž๐œ…๏ผ1(๐‘ž๏ผ1), and ๐”ฎ is a prime ideal of the first degree. If ๐‘ is any rational prime distinct from ๐‘”, and ๐‘“ the least exponent such that ๐‘๐‘“โ‰ก1 (mod. ๐‘š), ๐‘“ will be a factor of ๐œ‡, and putting ๐œ‡/๐‘“๏ผ๐‘’, we have ๐‘๏ผ๐”ญโ‚๐”ญโ‚‚ . . .๐”ญ๐‘’, where ๐”ญโ‚, ๐”ญโ‚‚ . . . ๐”ญ๐‘’ are different prime ideals each of the ๐‘“th degree. There are similar theorems for the case when ๐‘š 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 ๐–ฑ(๐‘’2๐œ‹๐‘–/๐‘), where ๐‘ is an odd prime; then this field is regular, and ๐‘ is said to be a regular prime,

when โ„Ž, the number of ideal classes in the field, is not divisible by ๐‘. Kummer proved the very curious fact that ๐‘ is regular if, and only if, it is not a factor of the denominators of the first 1/2(๐‘๏ผ3) Bernoullian