This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ... these congruences. The integers of the first class are the biquadratic residues of ir, nirl-1 for a ' = 1, modir, is the necessary and sufficient condition that a shall be a biquadratic residue of ir. The integers of the first and third classes are together the quadratic residues of ir, for they are the roots of the ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ... these congruences. The integers of the first class are the biquadratic residues of ir, nirl-1 for a ' = 1, modir, is the necessary and sufficient condition that a shall be a biquadratic residue of ir. The integers of the first and third classes are together the quadratic residues of ir, for they are the roots of the congruence 1 It is easily seen that every integer not divisible by I + - is a biquadratic residue of 1 + --. The integers of the second and fourth classes are together the quadratic non-residues of it, for they are the roots of the congruence a 2 =--1, modrr. The exponent of the power of i for which the congruence a 4 ==tr, mod-, r = o, 1,2,3 is satisfied is called the biquadratic character of a with respect to it and this power of i is denoted by the symbol (a/ir)4, so that we have always The symbol (at/ir), which is due to H. J. S. Smith, seems preferable to ((-/')), which was adopted by Jacobi, as by a change of subscript it will serve for the theory of residues of other degrees. If now (a/ir) have the meaning previously assigned, we see easily that modi If we understand by the quadratic character of o, mod f, instead of i or--i, the exponent of the lowest power of--I to which a is congruent, mod f, the notation for quadratic residues will be brought into accordance with that given above for biquadratic residues. The symbol (a/ir)4 obeys the following laws: From aqa, mod it, it follows that (-), -(3). (3), Since every integer a can be written in the form a = ir(i+iyPlp2..-P, where r = 0, i,2,3; s = o, or a positive integer; and pl, p2, .,"pn are odd primary primes, we have w4 = w, r )1 w, w, '' w; and the determination of the value of (-I is seen to be resolved w- into the determination of the values of...
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Seller's Description:
Very Good- No Dust Jacket Present. 8vo-over 7¾"-9¾" tall. 454 pages. "It has been my endeavor in this book to lead by easy stages a reader, entirely unacquainted with the subject, to an appreciation of some of the fundamental conceptions in the general theory of algebraic numbers. With this object in view, I have treated the theory of rational integers more in the manner of the general theory than is usual, and have emphasized those properties of these integers which find their analogues in the general theory." VERY GOOD-HARDCOVER, red cloth covers, lettering is bright on the spine.
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Seller's Description:
Very Good. Hardback book in very good condition, originally printed in 1910. reprinted 1926, and then this 1946 reprint of the mathematics classic, introduction by david hilbert.
As a graduate student, I studied algebraic number theory from "The Theory of Algebraic Numbers" by Harry Pollard. This is an excellent little monograph, but I felt the need for more details. I found them in "The Elements of the Theory of Algebraic Numbers" by Legh Wilber Reid. Reid presents all the details in the development of several specific algebraic number fields. It is an excellent book that I would recommend to any student wishing to understand algebraic number theory. It is easy to see why it has been reprinted in paperback form.
