Math II (Несколько текстов для зачёта), страница 8

Videla [11] has shown that the C[sub 2]-constructible regular polygons have p = 2[sup n]3[sup m]p[sub 1]p[sub 2] ellipse p[sub k] sides, where m, n, k >/= 0 and p[sub i] are distinct prime numbers of the form 2[sup a]3[sup b] + 1. Up to 300 sides, this corresponds to the 130 values 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, and 296.

If R[sub p] is known, R[sub 2p] can be RC-constructed from it by bisection of the sides, and R[sub 3p] can be C[sub 2]-constructed from it by trisection of the angles. If R[sub p] and R[sub q] are known, their superposition generates at least one side of R[sub m] where m is the least common multiple of p and q; this side can be replicated to obtain R[sub m].

Consequently, we have only to give RC-constructions for prime numbers of the form 2[sup a] + 1, which are 3, 5, 17, 257, 65, 537, .... There is very little probability that there exists another prime number of this form. For 2[sup a] + 1 to be prime, a must be a power of 2. Numbers of the form 2[sup 2][sup a] + i are called Fermat numbers. In June 1998, the smallest Fermat number 2[sup 2][sup a] + 1 not yet checked for primality was 2[sup 2][sup 24] + 1 [7]; this number has more than 5 million digits.

Similarly, we have only to give C[sub 2]-constructions for prime numbers of the form 2[sup a]3[sup b] + 1, with b > 0; there are 36 values up to 10[sup 6]: 7, 13, 19, 37, 73, 97, 109, 163, 193, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10,369, 12,289, 17,497, 18,433, 39,367, 52,489, 139,969, 147,457, 209,953, 331,777, 472,393, 629,857, 746,497, 786,433, 839,809, 995,329. There are 8 more candidates in [10[sup 6], 10[sup 7]], 8 in [10[sup 7], 10[sup 8]], 7 in [10[sup 8], 10[sup 9]], 7 in [10[sup 9], 10[sup 10]], and a total of 231 values in [1, 10[sup 30]].

Constructions of R[sub 2], R[sub 3], and R[sub 5] (and, consequently, R[sub n] for n = 2[sup k], n =3 Center dot 2[sup k], n = 5 Center dot 2[sup k], n = 15 Center dot 2[sup k]) were known in antiquity. For larger polygons, the only geometric constructions known are transpositions of algebraic solutions.

The complex numbers corresponding to the vertices of R[sub n] are zeros of the polynomial Z[sup n] - 1. For n an odd prime, the irreducible factors in Q[Z] of this polynomial are Z 1 and P[sub n] = Z[sup n - 1] + Z[sup n - 2] + ellipse + Z + 1.

The idea of the constructions is to decompose the extension of Q by the zeros of P[sub n] into successive extensions of degrees 2 and 3, whose elements can be obtained by C[sub 2]-constructions from the elements of the previous extension, starting with rationals. We will illustrate this in the following paragraphs. For the definition of field extensions, Galois groups, and their connection with geometric constructions, see [10].

P[sub n] has no real zero and (n - 1)/2 pairs of conjugate zeros. The monic polynomial Q[sub n] whose zeros are 2 cos(2kpi/n) has degree (n -1)/2 and integer coefficients. It is defined by the equation Z[sup (n-1)/2>Q[sub n](Z + 1/Z) = P[sub n]. It is easier to construct the zeros of Q[sub n]. In that case, the polygon obtained will be inscribed in the circle of radius 2 centered at the origin. Unless specified, this will be the case in all the following constructions.

Gauss has given in [6] an efficient algorithm to build the sequence of equations defining the zeros of Q[sub n]. In the following paragraphs, we try to introduce this algorithm step by step, with some simplifications proposed later in the literature.

R[sub 5]

Q[sub 5] = X[sup 2] + X - 1 has zeros (-1 +/- Square root of 5)/2. These zeros can be constructed with the Carlyle algorithm, as shown in Figure 5. The Carlyle circle has diameter [JB] and center L, with J(0, 1), B(-1, -1), and L(-1/2, 0).

R[sub 7]

We can construct the zeros of Q[sub 7] = X[sup 3] + X[sup 2] - 2X - 1 using one of the methods in the previous section. Figure 6 shows a construction using the method of Figure 2.

R[sub 9]

Although 9 is not a prime and can be constructed from R[sub 3] by trisection, we can give a simple construction of it (Fig. 7). Q[sub 9] = (X + 1)(X[sup 3] - 3X + 1). The zero -1 corresponds to the nontrivial vertices of R[sub 3]. The three other zeros can be obtained by intersecting the hyperbola XY = 1 and the parabola Y[sup 2] + X - 3Y = 0. The axis of this parabola is Y = 3/2, and it contains the points (-4, -1), (2, 1), and (0, 0) (dark dots).

R[sub 13]

With 13 sides, interesting things begin to happen. Q[sub 13] = X[sup 6] + X[sup 5] - 5X[sup 4] - 4X[sup 3] + 6X[sup 2] + 3X - 1 has degree 6. As we can only construct zeros of polynomials of degree 2 and 3, we have two choices:

1. Arrange the six zeros of Q[sub 13] in three groups of two

values, each pair being the zeros of a polynomial of degree

2. In that case, the coefficients of the polynomials of

degree 2 are in an extension of Q of degree 3.

2. Arrange the six zeros of Q[sub 13] in two groups of three

values, each triple being the zeros of a polynomial of degree

3. In that case, the coefficients of the polynomials of

degree 3 are in an extension of Q of degree 2.

Three groups of two values There are 15 ways of grouping the 6 zeros x[sub k] of Q[sub 13] into 3 pairs, as enumerated in the following array [the pair (x[sub i] x[sub j]) is noted i.j]:

1.2, 3.4, 5.6; 1.2, 3.5, 4.6; 1.2, 3.6, 4.5

1.3, 2.4, 5.6; 1.3, 2.5, 4.6; 1.3, 2.6, 4.5

1.4, 2.3, 5.6; 1.4, 2.5, 3.6; 1.4, 2.6, 3.5

1.5, 2.3, 4.6; 1.5, 2.4, 3.6; 1.5, 2.6, 3.4

1.6, 2.3, 4.5; 1.6, 2.4, 3.5; 1.6, 2.5, 3.4

Let omega be a primitive root of 1; in this array, k represents the zero x[sub k] = omega[sup k] + omega[sup 13-k] of Q[sub 13]. If we consider, for example, the pair (x[sub 1], x[sub 3]) noted 1.3, its two values are zeros of X[sub 2] - sX + p, with s = x[sub 1] + x[sub 3] = omega + omega[sup 3] + omega[sup 10] + omega[sup 12] and p = x[sub 1]x[sub 3] = omega[sup 2] + omega[sup 4] + omega[sup 9] + omega[sup 11]. By taking the successive powers of s, reduced modulo P[sub 13](omega), and looking for vanishing rational linear combinations of these powers, we can find a polynomial of minimal degree with rational coefficients having s as a zero: X[sup 6] + 2X[sup 5] - 7X[sup 4] - 6X[sup 3] + 5X[sup 2] + 5X + 1.

Let theta = e[sup 2pii/13] When omega takes the 12 values {theta, theta[sub 2], ..., theta[sup 12]} of the primitive 13th roots of 1, x[sub 1] + x[sub 3] takes only 6 values. These values are the zeros of the polynomial we obtained.

Consequently, we have to select the pairs taking only 3 values of s when omega takes all its 12 possible values. We can restrict our search to the five pairs 1.k for k = 2, 3, 4, 5, 6 (because the other pairs are obtained from these when omega varies). It appears that only the pair 1.5 takes three values of s, which are theta[sup 12] + theta[sup 8] + theta[sup 5] + theta for omega is an element of {theta, theta[sup 5], theta[sup 8], theta[sup 12]}, theta[sup 11] + theta[sup 10] + theta[sup 3] + theta[sup 2] for w *[This character cannot be converted to ASCII text] {theta[sup 2], theta[sup 3], theta[sup 10], theta[sup 11]}, and theta[sup 9] + theta[sup 7] + theta[sup 6] + theta[sup 4] for omega is an element of {theta[sup 4], theta[sup 6], theta[sup 7], theta[sup 9]}. The pairs corresponding to the last two values of s are, respectively, 2.3 and 4.6. The polynomial having these three values of s as zeros is H = X[sup 3] + X[sup 2] - 4X + 1.

We see that among the 15 possible choices, only 1.5, 2.3, 4.6 meets our needs. We can factor Q[sub 13] in the extension of Q by the zeros of H. Using Maple, one would type

> alias(alpha = RootOf(_Z^3 + _Z^2 - 4*_Z + l)):

# This is H

> factor(X^6 + X^5 - 5*X^4 - 4*X^3 + 6*X^2 + 3*X - l, alpha);

# This is Q_13

(X[sup 2] - alphaX + alpha[sup 2] + alpha -3)

(X[sup 2] + (alpha[sup 2] + 2alpha -2)X + alpha,

(X[sup 2] + (-alpha[sup 2] - alpha + 3)X

- alpha[sup 2] - 2alpha + 2)

The last two factors are obtained from the first one (X[sup 2] - alphaX + alpha[sup 2] + alpha - 3) when a takes its three possible values (the zeros of H).

We can now construct R[sub 13] (Fig. 8). First, we construct the zeros of H using the intersection of the hyperbola XY = 1 and the parabola p[sub 1]: Y[sup 2] + X - 4Y + 1 = 0 (dark gray). This parabola has axis Y = 2 (dark gray dashed line) and contains the points (2, 1), (-1, 0), and (3, 2) (dark gray points).

For each of the resulting values of alpha, we have to compute the zeros of X[sup 2] - alphaX + alpha[sup 2] + alpha - 3; that is, X[sup 2] - sX + p with s = alpha and p = alpha[sup 2] + alpha - 3.

For this, we construct the parabola p[sub 2]: Y = X[sup 2] + X - 3 (light gray), giving Y = p from X = s.p[sub 2] has axis X = - 1/2 and contains (0, -3), (1, -1), and (2, 3) (light gray points). Using this parabola, we obtain the points (s, p = s[sup 2] + s - 3) used in Carlyle's method. The three Carlyle circles (lighter gray) give the six zeros of Q[sub 13] (lighter gray vertical dashed lines).

Two groups of three values As we have seen earlier, we have to find a triple 1.u.v of zeros whose associated sum x[sub 1] + X[sub u] + X[sub v] = omega + omega[sup 12] + omega[sup u] + omega[sup 13 - u] + omega[sup v] + omega[sup 13 - v] takes only two values when omega takes all the values theta[sup k] for k = 1, 2, ..., 12. After computations, it appears that only the triple 1.3.4 satisfies this criterion. The two values it takes are the zeros of H = X[sup 2] + X - 3. One factor of Q[sub 13] in the extension of Q by the zeros of H is

X[sup 3] - alphaX[sup 2] - X - 1 + alpha,

where alpha is an arbitrary zero of H (taking the other zero gives the other factor).

The construction of R[sub 13] (Fig. 9) deduced from this decomposition begins with the construction of the two zeros of H using the Carlyle circle centered at (- 1/2, - 1) and containing J(0, 1).

For each zero alpha of H, we then construct the three zeros of X[sup 3] - alphaX[sup 2] - X - 1 + alpha using the intersection of the hyperbola XY = 1 and the parabola (-1 + alpha)Y[sup 2] + X - Y alpha = 0. This parabola contains the points (0, - 1), (0, alpha + 3), (2, 1), (2, alpha + 1), (alpha, 0), and (alpha, alpha + 2). The points (0, - 1) and (2, 1) are shared by the two parabolas and are shown in black in Figure 9; the other points used in the construction of the parabolas are filled dots.

R[sub 17]

Gauss established the RC-constructibility of R[sub 17] in 1796 (see [1, 4, 5, 9] for historical details about this discovery and other constructions proposed afterward).

As we have seen for R[sub 13], the construction is obtained by finding the zeros of three successive polynomials of degree 2, the coefficients of a polynomial being constructed from the zeros of the previous polynomial.

We have Q[sub 17] = X[sup 8] + X[sup 7] - 7X[sup 6] - 6X[sup 5] + 15X[sup 4] + 10X[sup 3] 10X[sup 2] - 4X + 1. Only the pair 1.4 take four values: 1.4, 2.8, 3.5, and 6.7. They are the four zeros of H[sub 2] = X[sup 4] + X[sup 3] 6X[sup 2] - X + 1. Continuing to the next level, we look for a pair of these pairs taking only two values. The only choices are 1.2.4.8 and 3.5.6.7, the two zeros of H[sub 1] = X[sup 2] + X - 4. Let alpha be a zero of H[sub 1], we can factor H[sub 2] as H[sub 2] = (X[sup 2] + X + alphaX - 1)(X[sup 2] - alphaX - 1). Let beta be a zero of X[sup 2] - alphaX - 1; Q[sub 17] can be factored in Q[beta], one of the factors being X[sup 2] - betaX 3/2 + beta/2 + alphabeta/2 - alpha/2 [the four factors are obtained by substituting the four possible values of (alpha,beta) in this factor].

In the construction (Fig. 10), the values of alpha (1.2.4.8 and 3.5.6.7) are obtained using the circle centered at (- 1/2, -3/2) containing J(0, 1) (dark gray). The values of beta (1.4, 2.8, 3.5, and 6.7) are constructed using two circles, centered at (alpha/2, 0) and passing through J (medium gray). The eight roots of Q[sub 17] are obtained by four circles (light gray). To simplify, instead of constructing -3/2 + beta/2 + alphabeta/2 - alpha/2, we can remark that x[sub 1]x[sub 4] = x[sub 3] + x[sub 5], x[sub 2]x[sub 8] = x[sub 6] + x[sub 7], x[sub 3]x[sub 5] = x[sub 2] + x[sub 8], and x[sub 6]x[sub 7] = x[sub 1] + x[sub 4]. Gauss used sign tests to match the sums with the different roots. We have used numerical approximations.

R[sub 19]

Q[sub 19] = X[sup 9] + X[sup 8] - 8X[sup 7] - 7X[sup 6] + 21X[sup 5] + 15X[sup 4] - 20X[sup 3] - 10X[sup 2] + 5X + 1. As in the case of 17 sides (8 = 2 Center dot 2 Center dot 2), we do not have a choice of the decomposition (9 = 3 Center dot 3). We have to find the zeros of a third-degree polynomial, and then the zeros of other third-degree polynomials whose coefficients depend on the first three zeros.

Gauss proposed a way to find directly the suitable triples or pairs, which we did by systematic search in the last paragraphs. We first find a number p in 1, 2, ..., 18 whose successive powers modulo 19 generate a permutation of 1, 2, ..., 18; for example, p = 2 can be chosen because the successive powers of 2 modulo 19 are 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10.