[glashan] reduction OF CIRCULANTS ' 45 



+0n'S'59j]-\-2[0n'3^Q8j-^0n'4'Q79 + 0nW2S4] 

 -31[œp42236+œP52479]+35[0n^4--=39j + œP5-28/-] 

 - 73[œP23697-] + 15[0n^2378j] - 18[œP2468j] 

 -29[œP24789]+37[œP2567i]+81[œP25689] 

 + 70[0'''P^34689] - 7[0n345678] - 15[0n ^2^325^] 



-4[0-n"229n0^] + 7[0n-^3282/] + 36[œP2^3--^46 + œP227258] 



+ 3[0n-2H235]^g9jQ3p2^'4%-]-30]0n2225^89 + 0n -2^/34] 



-S[0'V2'-5'^7j-h0nW5'^Q9 + 0n''3^^f79]-52[0n'~2^7'3j] 



+ 58[03P227249 + on -'3^8227] - 1 9 [0^^22923 6 +0n '•'324^77] 



+ 14[œP228^47 + œP228256 + œP229245 + 0'n2325278] 



+ 25[œP324289]+47[0=^P527^34]-41[œP527%-+œP328245] 

 + 6[œP2'359j] + 28[œP2-368j]-38[œP224689] 



-16[0'n22-3789+0-'P3=^456; + 0n-5^679j + 0n272468j] 

 + 61[œP2^4587+0n^3-4579 + 0n2/2578] + 94[œP225679] 

 +83[0n222467j^03P5^2478]+39[œP3^259j] + 116[œP3^258j] 

 -5[œP32267j+0n^592469 + œP7-'2348 + œP7^5689] 

 + 17[œP322689 + œP-52236j + 0n35-^2379+œP7^'459j+œP/3568] 

 -71[œP324678 + œPr369j + œP/2389] + 50[œP5'^3468] 

 -60[œP732456j - 126[œP/2479] -27[œP/2569] 

 + 72[œP/3478+œP/-4567]-98[œP23457j] 



- 109[œr->234589] +45[œP234679] - 65[œP235678] 

 + 100[œP'26789i] + 12[œP35789i] + 78[0n-^45689i] 

 + 57[0n234789j] - 23[œP2232629] - 56[0-P2^43529] 

 + 87[0n22242725] + 141[œP2-3247j]+9[0n22333489 

 + 0n22242367]-112[0-P2532579]+31f02r^2^3'^678] 



- 79[02P2242379 + 0^22^523(38] _2[02i2224a568] 



+ 75[œP225^34j] + 64[0n2225M67]-57[0n^324^25j] 

 -35[œP2242269] - 13[0n23243278] + 29[0n'^2234569] 



- 59[02p2234578+02P3-4679j] - 70[02p3'-24568] 



- 136[0-P322789j] + 106[œP324589j] -26[œP-325678j] 

 + 190[0212234689i] - 96[^P235679j] + 80[0"n=5245678j] 



- 63[0n23456789] - 61510123456789J] } . 



The importance of the circulants lies in the fact that the question 

 of the solution by radicals (or circula^ functions) of the equation 



C(w) =0 is reducible to the problem of determining 3/1", 3^2'* yll^i 



explicitly in terms of C2, Cz,... .c„, i.e., the solution by radicals" of 

 Q(w)=x«+g2A;"-2+ç3.r'^-3. ...g„ = is the inverse of the problem 

 considered in this paper. This was proved by Abel and his demon- 

 stration that the general equation of degree higher than four is not 

 resolvable in terms of radicals alone, rests on his identification of 

 Q(w) and C(«). 



