Opera Omnia: Introductio In Analysin Infinitorum by Euleri L.

By Euleri L.

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Q = e 2~iz. Then, by II w To see the behavior of ~ at co, put j(z) = q'l(l + A(q)) where A(q) is a power series of q with integral rational coefficients and A(O) = O. Let a s Ms ~ as b 0 d ' e2~iMs(Z) =~d bs d qS 2~i , s Then, a a -b s js(z) -- J(Ms(Z)) - - S d s = e s S (2) d --- q- ds S bd (1 + A ( ~ d s 5) ) . d s Iii-5 Therefore, ~ can only have the singularity of a pole in q at q = O, and, hence, is a polynomial in j over C. ~ Furthermore, the coefficients ~fithe q-expansion of are all algebraic integers in the field q ( ~ n ), ~ n Galois automorphism of Q(Sn)/Q.

In a class field L over K if any only if p divides the conductor f(L/K) of L/K. Before going to state the next theorem, Artin's reciprocity law, we first make some preparations. of Let L/K be an arbitrary Galois extension degree n with the Galois gro%o G. Let ~ be a prime ideal of K unramified in L, and [ a prime ideal of L dividing ~. Then there exists a unique element CT in G such that (4) ~(~) ~ = NK/Q(~) mod. P I for any algebraic integer ~ in L. This element ~ris called the Frobenius substitution of P_ for the extension L/K and is denoted by O-p; it generates I the so-called decomposition group of the prime ideal P, aad if f is the order of ~ p and n = fg, then I -PI = P ' where Pi are distinct prime ideals of L.

P P Let now qo " exp'(2"i~I/m2)" The numbers J(qo), ji(qo), ~i(qo) are algebraic integers. Since Gp(Jq,u,Ji, ~j) may be ~rritten as a polynomial in t and j with coefficients in pZ, its value for q = qo is a polynomial in u, whose coefficients are algebraic integers divisible by p. (n) Thus ap(j(a) p, u, Ji(a), ~k(%, ~2)) TO rood p. Let us now put u = ~ ( ~ , =2 ) = ~ d ( ~ , =2 ). Then by (ii) and the definition of % ~ we get (12) (0(a)p - Sd(a_)) 7T (@d(~, ~2) - ~i(%, ~2)) -= 0 ~d p. Iv-9 Since ~divides (13) (p), we have afortiori (j(a)p - jd(z)) i7r ~ (~d (5, =2 ) " ~ i ( 5 ' ~2)) ----0 ~ d ~" By Theorem I, , a2) mod 2 and the right hand side generates the ideal ~12, which is prime to p.

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