LINDEMANN WEIERSTRASS THEOREM PDF

A more elementary proof that e is transcendental is outlined in the article on transcendental numbers. Modular conjecture An analogue of the theorem involving the modular function j was conjectured by Daniel Bertrand in , and remains an open problem. Let q1, If a1,

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A more elementary proof that e is transcendental is outlined in the article on transcendental numbers. Modular conjecture An analogue of the theorem involving the modular function j was conjectured by Daniel Bertrand in , and remains an open problem. Let q1, If a1, To simplify the notation set: Then the statement becomes: Let p be a prime number and define the following polynomials: where l is a non-zero integer such that are all algebraic integers.

Define: [7] Using integration by parts we arrive at: where , and is the j-th derivative of. This also holds for s complex in this case the integral has to be intended as a contour integral, for example along the straight segment from 0 to s because is a primitive of.

Let us consider the following sum: In the last line we assumed that the conclusion of the Lemma is false. In order to complete the proof we need to reach a contradiction.

We will do so by estimating in two different ways. Now Since each is obtained by dividing a fixed polynomial with integer coefficients by , it is of the form where is a polynomial with integer coefficients independent of i. The same holds for the derivatives. Hence, by the fundamental theorem of symmetric polynomials, is a fixed polynomial with integer coefficients evaluated in this is seen by grouping the same powers of appearing in the expansion and using the fact that are a complete set of conjugates.

So the same is true of , i. This proves Lemma A. Lemma B. The product vanishes by assumption, but by expanding it we obtain a sum of terms of the form multiplied by integer coefficients. Since the product is symmetric, we have that, for any , has the same coefficient as. Thus after having grouped the terms with the same exponent we see that the set of the exponents form a complete set of conjugates and, if two terms have conjugate exponents, they are multiplied by the same coefficient.

So we are in the situation of Lemma A. To reach a contradiction it suffices to see that at least one of the coefficients is non-zero. This is seen by equipping C with the lexicographic order and by choosing for each factor in the product the term with non-zero coefficient which has maximum exponent according to this ordering: the product of these terms has non-zero coefficient in the expansion and does not get simplified by any other term.

This proves Lemma B. Final step We turn now to prove the theorem: Let a 1 , Then let us assume that: We will show that this leads to contradiction and thus prove the theorem. Let us denote the roots of this polynomial a i 1, Then according to our assumption: where the product is over all possible choices. Since the product is over all possible choices, each of b 1 , Each of the latter is a rational number as in the proof of Lemma B. Thus b 1 , Therefore, according to Lemma B, the equality cannot hold, and we are led to a contradiction which completes the proof.

Paris, 77, pages 18—24, Boron, Dover Publications, The rest of the proof of the Lemma is analog to that proof. Further reading.

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In order to complete the proof we need to reach a contradiction. This proves Lemma A. Lemma B. Thus after having grouped the terms with the same exponent we see that the set of the exponents form a complete set of conjugates and, if two terms have conjugate exponents, they are multiplied by the same coefficient. So we are in the situation of Lemma A.

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