“About a property of the epitome of all real algebraic numbers. (By Mr. Ganter in Hall a. S.)
A real algebraic number is generally understood as a real number size w, which is sufficient for a non-identical equation of the form: (1.) ao to* a, w"--1--1-. • • a„ = 0, where n, ao, ch, • • • a. integers are; we can think of the numbers n and ao positively, the coefficients ao, ao, without community 'healer and the.Equation (1.) irreductibly; these determinations are achieved that according to the well-known principles of arithmetic and algebra, the equation (1.), which a real algebraic number is sufficient, is a completely determined; vice versa, as you know, belong to an equation of the form (1.) at most as many real algebraic numbers w, which are sufficient for her as you give degrees n. The real algebraic numbers form in their entirety an epitome of number sizes, which is called (w); it has the same as can be seen from simple considerations, such a texture that there are an infinite number of numbers from (w) in every vicinity of a thought-all the more striking - this should therefore be the remark for the first sight that the epi Certain real algebraic number w belongs, so, in other words, the epitome (w) in the form of an infinite legal series: (2.) CO" CO23 • • Wv3 "
Cantor, on the theory of algebraic numbers.
259 can be thought of, in which all individuals of (w) occur and each of them is in a certain place in (2nd), which is given by the corresponding index. As soon as you
Cantor, on the theory of algebraic numbers. 259 can be thought of in which all individuals of (w) occur and each of them is in a certain place in (2nd), which is given by the corresponding index. As soon as you have found a law according to which such an assignment can be thought of, the same can be modified according to arbitrariness; it will therefore suffice if I use the least circumstances in §. 1, which seems to me to take up the fewest circumstances. In order to give an application from this property of the epitome of all real algebraic numbers, I add §. 2 to §. 1, in which I show that if any series of real number sizes of the form (2.) is present, you can sing numbers /7" at any predetermined intervals (a • • • (3) that are not included in Furthermore, the sentence in §. 2 presents itself as the reason why epitomages of real number sizes that form a so-called continuum (such as all real numbers, which are 0 and 1) cannot be clearly related to the epitome (v); so I found the clear difference between a so-called continuum and an epitome of the type of
§. 1. If we go back to the equation (1.), which an algebraic number w is sufficient and which, according to the intended determinations, is a completely specific one, may the sum of the absolute amounts of their coefficients, increased by the number n-1, where n indicates the degree of w, be called the height of the number w and designated with N; it is therefore N = n-1 -I-. [a0] [a,] • • • -1-[4. The height N is therefore a certain positive integer for each real algebraic number w; conversely, there is only a finite number of algebraic real numbers with the height N for each positive integer of N; the number of the same is 9(N); it is example-33.
260 C.antor, on the theory of algebraic numbers. -wise 9(1) 1; cp(2) = 2; cp(3) .4. The numbers of the epitome (w), i.e. all algebraic real numbers can then be arranged as follows masses; take as the first number o), which is a number with the height .1■7 = 1; let the cp(2) = 2 algebra
260 Gastor, between Theory of algebraic numbers. -wise 9(1) 1; (p(2) = 2; (p(3) = 4. The numbers of the epitome (w), i.e. all algebraic real numbers can then be arranged as follows masses; one takes as the first number ah the one number with the height N ■ 7 = 1; let them, rising in size, which (p(2) = 2 alge-braic real numbers with the height N = 2 follow, they Place are shown, the real alge-braic numbers with the height N = Ni -1- 1 follow them and ascending in size; thus one obtains the epitome (w) of all real alge-braic numbers in the form: ah, w2, 04, and can speak of the yten real alge-braic number with regard to this arrangement,
§. 2. If there is an infinite row of different real number sizes (4.) ah, ah, • • • (4, • • • • • • • • • •, given according to any law, a number 1i (and therefore an infinite number of such numbers) can be determined at any predetermined interval (a • • • (3), which does not occur in the row (4th); this is now to be proven. We assume at the end of the intervals. (a • • • (3), which is given to us arbitrarily, and it is a < ß; the first two numbers of our series (4th), which lie inside this interval (with exclusion of the limit,ry, may be designated with a', ß', and it is a' < (3'; So here are a, a" • • • by definition certain. Numbers of our series (4.),,.whose "indidices are in continuous rise, and that the same applies to the numbers ß', (3"•• • • ; further take .the numbers a, a",-.. •
Cantor, on the theory of algebraic numbers.
In terms of their size, the numbers 11, (3" • continuously decrease in size; from the intervals (a • • • ß), (a' • • • 13'), (a" • • • fi"), • • • each of them all includes the same following. — Two cases are now conceivable here.
Cantor, , on the theory of algebraic Zahlest. 261
Continuously increasing in their size, the numbers /1, (3", • continuously decrease in size; from the intervals (a • • • /3), (a' • • • (1), (a" • • • /3"), • • • each one includes all of the same following. — Two cases are now conceivable here. Either the number of intervals thus formed is finite; the last of them is (ao • • • (3'")); since there can be at most one number of the row (4th) inside it, a number •7 can be assumed in these intervals, which is not contained in (4th), and the sentence is thus proven for this case. Or the number of intervals formed is infinite; then the variables a, ä, a", • • because they continuously increase in size without growing infinity, have a certain limit value h cec°; the same applies to the quantities (3, (3', (3", because they continuously decrease in size, their limit value is (1x) ; if ac° =ß ( Can*); but is at < (3°°, so is any number sufficient? I inside the interval (a°° • • • (122) or also at the limits of the same of the demand not to be included in the series (4th). -The sentences proven in this essay allow extensions in different directions, of which only one is mentioned here: "Is oh, oh, • • • • • w,,, • • • • • • a finite Of those numbers 12 that can be represented as rational functions with integer coefficients from the given numbers co, there are infinite numbers at each interval (a • • • (3) that are not contained in (S2)." In fact, one convinces oneself by a similar conclusion,
•) If the number 77 were included in our series, you would have 27W," where p is a certain index; but this is not possible, because to, is not inside the interval Vt(P)... A0), while by definition the number 37 has this interval inside.
262 Castor, sur nierie der akjebraieehen numbers.'
As in §. 1, that the epitome (s2) can be understood in the series form: 1219 S211 • • • 12., •,•; from which, with regard to this §. 2, the correctness of the sentence follows. A very specific case of the sentence mentioned here (in which
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262 Cantor, on the theory of algebraic numbers.
As in 1, that the epitome (S2) can be understood in the series form: S21, S22, • • • 22., • • •; from which, with regard to this §. 2, the correctness of the sentence follows. A very specific case of the sentence mentioned here (in which the series • w" • • • is a finite and the degree of rational functions that provide the epitome (d2) is a predetermined), with reference to Galois Principien, by Mr. B. Minnigerode has been proven. (See Math. Annals by CIAsch and Neumann, vol. IV. S. 497.)
Berlin, the 23rd December 1873.