Theorie der algebraischen Zahlen

Abstract

IN this volume Dr. Hensel gives the first instalment of a treatise on algebraic numbers, embodying an independent method on which he has been engaged for the last eighteen years. Its leading idea may be illustrated by the following example. Let us take the solvable congruence, x² ≡ 2 (mod. 7), the roots of which are x ≡ 3 and x ≡ 4. The same congruence can be solved with respect to the moduli 7², 7³, 7⁴, and we obtain the solutions, in least positive residues, (3, 4), (10, 39), (108, 235), (2116, 285), and so on. Taking the first number in each bracket and expressing it in the septenary scale, only writing the digits in the reverse of the usual order, we obtain the associated solutions, 3, 31, 312, 3126; and it is clear that if x = a₁a₂... a_n, is a solution of x₂ ≡ 2 (mod. 7ⁿ), we can find a number a₁a₂... a_na_n+1, which is a solution of x² ≡ 2 (mod. 7^{n+ 1}). There is thus a definite sequence of digits, 3, 1, 2, 6,... a_n,... such that each is a least positive residue of 7 (or zero), and such that 3126... a_n, is a solution of x² ≡ 2 (mod. 7ⁿ). This sequence niay be said to be the symbolical septenary representation of √2. But conversely we may take any such sequence, a₁a₂... a_n... and define it as a septenary number, in an extended sense. All such numbers form a corpus, provided we introduce septenary fractions of the same type. Since - 1 ≡ (7ⁿ - 1) (mod. 7ⁿ), the symbolical form of - 1 is 666... or 6; hence every ordinary positive or negative integer or fraction has a symbolic expression which is wholly or partly periodic, e.g. 2/3 = (3+6)/3=32, and so on. Similar results hold for any prime modulus; when the modulus is composite, some curious anomalies occur.

Theorie der algebraischen Zahlen.

By Dr. Kurt Hensel. Erster Band. Pp. xii + 350. (Berlin and Leipzig: Teubner, 1908.) Price 14 marks.