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 nl, 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 ax-{-By of which any given ideal is the content.

58. Now let w1, w1, w, . be a basis of o; u1, 141, . un a set of indeterminate; and

E=w1ui+wm+ - - - +w»u, .:

5 is called the fundamental form of 9. It satisfies the equation N (x-5) =O, or

F(x) =x"+U1x""+ . . -l-U, .=o

where U1, U2, . Un are rational Eolynomials in 141, ug, .un with rational integral coefficients. T is is called the fundamental equation.

Suppose now that p is a rational prime, and that p=p“q"r°. . 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 ide ntical congruence

F(x) ={P(x)}“{Q(x)}”{R(x)l°- - -(m0d 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'u1"'. un will be congruent, mod p, to the corresponding coefficient in'F (x). Iff, g, h, &c., are the degrees of p, q, r, &c. (§ 47), thenf, g, h, are the dimensions in x, u1, u2, . . u.. of the forms of P, Q, R, respectively. For every prime p, which is not a factor of A, a=b=c=. =I 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 SZ, F(x) is a prime function (mod p) and conversely.

It generally happens that rational integral values a1, aa, an can be assigned to u1, ug, . un such that Un, the last term in the fundamental equation, then has a value which is prime to p. Supposing that this condition is satisfied, let a1w1+rL1w1+ .+a, .w,1=a; and let P1(a) be the result of putting x=a, u¢=a; in P(x). Then the ideal p is completely determined as the greatest common divisor of p and P1(a); 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”'- I =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'<, a power of an odd prime. It will be convenient to writeu=¢(m)=g'<'I(q-I); if we also put r=e2ffi/M, the primitive roots of x"'= I will be, u in number, and represented by r, r", fb, &c. where I, 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) =o of degreep; the symbol f(x) denoting (x"'-1)-:~(x"'/'1—I). There are primitive roots of the congruence x»'=I (mod m); let g be any one of these. Then if we put r°h=r;, , we obtain all the roots of f(x) =o in a definite cyclical order (rr, rg .r, t); and the change of r into r” produces a cyclical permutation of the roots. It follows from this that every cyclic polynomial in 71, rg, .rn with rational coefficients is equal to a rational number. Thus if we write l+ag +bg'+.+kgu'1 =n, we have, in virtue of r;, =r”'°, r1“r1°...r, .~-1"1, ».'=r", and, if we use S to denote cyclical summation, S(r1“r1". JM) = r"+r""+. . +r”°"", the sum of the nth powers of all the roots of f(x) =o, and this is a rational integer or zero. Since every cyclic polynomial is the sum of parts similar to S(r1“r2b. . .r, i'), the theorem is proved. Now let e, f be any two conjugate factors of p, so that ef = il, and let

lli=7i'i'7i+¢'i"f1Z+2e"l"- . 'i"7i+(/-1)e (i=I, 21- - -6) then the elementary symmetric functions Em, Enmf, &c., are cyclical functions of the roots of f(x) =o and therefore have rational values which can be calculated: consequently 171, 112, . sq., which are called the f-nomial periods, are the roots of an equation F(11) ='fle'l'5171° 1'l"- - . +v.=0

with rational integral coefficients. This is irreducible, and defines a field of order e contained in the field defined by f(x) =o. Moreover, the change of r into r” alters ni into "li+1, and we have the theorem that any cyclical function of -41, 172, . . . n, is rational. Now let h, k be any conjugate factors of f and put

Zi =f¢+f¢+1»+ff+m+~ - ~7d+(f-h)c (i= I, 21 3») then § 1, § '1+, , fm.. .§ '1+(;, 1), will be the roots of an equation GK) = § "'-1n§ ”"'1+f2§ ""°+- » - -Hn =0»

the coefficients of which are expressible as rational polynomials in 171. Dividing h into two conjugate factors, we can deduce from G(§ ') =0 another period equation, the coefficients of which are rational polynomials in 111, § '1, and so on. By choosing for e, h, &c., the successive prime factors of n, 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 1'1-1 for its roots. Thus to take a very interesting historical case, let m = 17, so that p = 16 = 24, the equations are all quadratics, and if we take 3 as the primitive roct of 17, they are

1l”+17'4=0, s“'-'/it-I =0 s

2>~'-2s“>+(11§ -n+s“-3)=0, P2"'}P'l'I=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-l-ax+b=o 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 ¢(m) is a power of 2; in other words m =2' P where P is a product of different odd primes, each of which is of the form 2"+1.

Returning to the case m=g*, we shall call the chain of equations F(1;) =O, &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 pt 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 u. No other chain of auxiliaries having similar properties can be formecrcontaining fewer equations of a given prime degree p; a fact first stated by Gauss, to whom this theory is mainliy due. Thus if m=q'= we must have at least (fc-I) auxiliaries of or er q, and if q-I =2¢;lpB . . ., we must also have a quadratics, B 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) =o

60. When m is an odd prime p, there is another very interesting way of solvin the equation (xp-1) + (x-1) =o. As before let (r1, 12, . . r, , § be its roots arranged in a cycle by means of a primitive root of x1"lEI (mod p); and let e be a primitive root of ef'-1 = I. Also let

01 =71+€f2+62T3+ . . . +EP°27p 1

0k=f1+ék72+£2k73+ . . +€ kfp 1 =2, 3, . . so that 0;, is derived from 01 by changing e into ek. The cyclical permutation (r1, r2, .rp-1) applied to gk converts it into e"'0;, ; hence 0101./0;at1 is unaltered, and may be expressed as a rational, and therefore as an integral function of e. It is found by calculation that we may put

m= I

¢k(€) =@'= §)+ eindm-l-kind(p+1-m) [k=I, Q

0'=+1 m=a

while

0101.4 = -p.

In the exponents of h, (e) the indices are taken to the base g used to establish the cyclical order (11, rg . r, , 1). Multiplying together the (p—2) preceding equalities, the result is

01""= -P1//1(e)1,02(e) .1//p 3(e) =R(e)

where R(e) is a rational integral function of e the degree of which, in its reduced form, is less than ¢>(p- I). Let p be any one definite root of x1'°1=R(e), and put 01 =p: then since

ok .

§ ;:=Pi¢z- ~ -'pkwe

must take 0;¢=p"///MP2 , bk 1 =R;, (e)p", where R1, (¢) is a rational function of e, which we may suppose put into its reduced integral form; and finally, by addition of the equations which define 01, 01, &c.,

(;D- I)f1=n+Rz(f)P”-I-Ra(f)p3+- - ~ +Rp 2(¢)P""-If in this formula we change p into e"'p, and 11 into r;H.1, it still remains true. 4

It will be observed that this second mode of solution employs a Lagrangian resolvent 01; 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 Me) have many curious properties, which have been investigated by Jacobi, Cauchy and Kronecker

61. When m =q'<, the discriminant of the corresponding cyclotomic field is =hq', where)=q" '(f<g-x-I). The prime q is equal to q", where p =¢(m) =q"" (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 pf§ I (mod. m), f will be a factor of, u, and putting /.¢/f=e, we have p=p1p2 . p, ,, where pi, pg . . pe 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 prirne. 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”"/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 Hp-3) Bernoullian