From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Research of a root of unity Date: 29 Jan 2001 21:46:05 GMT Newsgroups: sci.math Summary: Possible algebraic relations among some roots of unity? In article <3A746101.DF94CC08@club-internet.fr>, Nicola Sottocornola wrote: >Is it possible to find a number a different by zero s.t. > >1) a is the real part of a root of unity >2) -1 < a < 1/2 >3) a/(a-1) is also the real part of a root of unity I don't think so; the problem is that the minimal polynomials of the expressions a/(a-1) 'look different' from the minimal polynomials of the expressions a. Suppose a is the real part of a primitive n-th root of unity, w . (n>2). Then a = (1/2)( w + 1/w ) . Now, 2a = w + wbar is an algebraic integer, and its minimal polynomial has degree exactly phi(n)/2 . (Proof: it is clear that [ Q(w) : Q( 2a ) ] is exactly 2.) So real-parts-of- primitive-nth-roots-of-unity are precisely the d=phi(n)/2 roots of a certain integer polynomial P_n = 2^d X^d + ... It's not hard to compute this polynomial from the n-th cyclotomic polynomial; for example since P_n(1) = 1 unless n is a power of a prime p, in which case P_n(1) = p. In particular, notice that dividing the lead coefficient of P_n by its value at X=1 -- an expression independent of any scaling or content of P_n -- will be either 2^d or 2^d / p. What about b = a/(a-1) ? Since a = b/(b-1) too, we have Q(a) = Q(b), so that a and b have minimal polynomials of the same degree. Since P_n( b/(b-1) ) = 0, it follows that P'(b) = 0 where P'(X) is the numerator of the rational function P_n( X / (X-1) ), i.e. P'(X) = P_n(X/(X-1)) (X-1)^d is the minimal polynomial of b. (If P_n(X) = \sum a_k X^k, this is P'(X) = \sum a_k X^k (X-1)^(d-k); The lead coefficient of P' is then clearly equal to P_n(1), and P'(1) = a_d.) In particular, dividing the lead coefficient of P' by its value at 1 gives 1/2^d or p/2^d. Comparing the last two paragraphs we see that the only occasions under which both a and b can be real parts of roots of unity are a few with d (and e ) really small. The only solutions seem to be {a=-1, b=1/2}, {a=1/2, b=-1}, and {a=0, b=0}. You have explicitly excluded these in your search. dave ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: Research of a root of unity Date: 30 Jan 2001 15:33:35 GMT Newsgroups: sci.math I responded to this article yesterday but I don't like proofs that have unnatural turns to them so let me try again. In article <3A746101.DF94CC08@club-internet.fr>, Nicola Sottocornola wrote: >Is it possible to find a number a different by zero s.t. > >1) a is the real part of a root of unity >2) -1 < a < 1/2 >3) a/(a-1) is also the real part of a root of unity The problem here is that the relationship between a and b=a/(a-1) is symmetric but not obviously so. It's better to write it in the form (a-1)(b-1)=1, which shows the symmetry and also suggests where to begin. If a is the real part of a primitive n-th root of unity z, then we compute a-1 = (z-1)^2/(2z). Then the norm (from K = Q(z) to Q) of a-1 is (N(1-z))^2/2^d where d = [K:Q] = 2^phi(n) (except for n=2, in which case N(z) = -1, not +1 !) Now, N(1-z), the product of the conjugates of 1-z, may be evaluated by substituting 1 into the minimal polynomial of z, which is the n-th cyclotomic polynomial P_n. It is known that for n>1 we have P_n(1) = 1 unless n is a power of a prime p, in which case P_n(1) = p. Thus the norm of a-1 is 1/2^d or p^2/2^d . We will show that norms of this form are usually inconsistent with a relation (a-1)(b-1)=1. We can do it, say, with size. Observe that p^2/2^d is no larger than p^2/2^(p-1) and with a bit of calculus we find this to be less than 1 when p is at least 7. We can likewise work out that the norm of a-1 is less than 1 even when n is a (slightly larger) power of 2, 3, or 5. In summary: the only exceptions to N(a-1) < 1 are when (n=1, n=2 or) n=3, n=4, and n=5. Technically, it is a bit easier to work with the absolute norm, N(x) = | N_{K/Q} ( x ) | ^ (1/[K:Q]) since this value is independent of the field K in which the algebraic number x lies. Clearly it is still true that 0 < N(a-1) < 1 except when n=1,2,3,4, or 5. In those cases we have N(a-1) equal, respectively, to 0, -2, 3/2, 1, and sqrt(5)/2. If then b is the real part of a primitive m-th root of unity, too then we likewise have 0 < N(b-1) < 1 unless m=1,2,3,4, or 5. This contradicts the assertion (a-1)(b-1) = 1 unless at least one of n and m is small, but these cases are easily dispensed with: If n=1, a=1, and (a-1)(b-1)=1 is impossible If n=2, a=-1 and so b=1/2, corresponding to m=6 If n=3, a=-1/2 and so b=1/3, not the real part of a root of unity If n=4, a=0 and so b=0, corresponding to m=4 If n=5, a=(sqrt(5)+-1)/4 making b=-1/sqrt(5) or -2-sqrt(5), in neither case the real part of a root of unity. (One may also dispose of the cases n=3 and n=5 using norm arguments as I did in the previous post.) So the only pairs {a,b} of real parts of roots of unity which are related by (a-1)(b-1)=1 are {-1,1/2} and {0,0}. dave