proof of Cauchy-Davenport theorem
There is a proof, essentially from here http://www.math.tau.ac.il/ nogaa/PDFS/annr3.pdf(Imre Ruzsa et al.):
Since is a field, there are polynomials of degree and of degree such that for all and for all . Define a polynomial by .
This polynomial coincides with for all in and in , for these , we have, however, . The polynomial is of degree in and of degree in . Let , then is zero for all , and all coefficients must be zero. Finally, is zero for all , and all coefficients of must be zero as elements of .
Should the assertion of the theorem be false, then there are numbers , with and and .
But the monomial does not appear in the second and third sum, because for we have , and for we have . Then is equal to , this is equal to the binomial coefficient , which is not divisible by for , a contradiction. The Cauchy-Davenport theorem is proved.
|Title||proof of Cauchy-Davenport theorem|
|Date of creation||2013-03-22 14:34:49|
|Last modified on||2013-03-22 14:34:49|
|Last modified by||Wolfgang (5320)|