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In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).
This pen-and-paper method uses the same algorithm as polynomial long division, but mental calculation is used to determine remainders. This requires less writing, and can therefore be a faster method once mastered. The division is at first written in a similar way as long multiplication with the dividend at the top, and the divisor below it.
Note: "lc" stands for the leading coefficient, the coefficient of the highest degree of the variable. This algorithm computes not only the greatest common divisor (the last non zero r i ), but also all the subresultant polynomials: The remainder r i is the (deg( r i −1 ) − 1) -th subresultant polynomial.
A polynomial P with coefficients in a UFD R is then said to be primitive if the only elements of R that divide all coefficients of P at once are the invertible elements of R; i.e., the gcd of the coefficients is one. Primitivity statement: If R is a UFD, then the set of primitive polynomials in R[X] is closed under multiplication.
Every univariate polynomial with real coefficients of degree larger than two has a factor of degree two with real coefficients. Every univariate polynomial with real coefficients of positive degree can be factored as c p 1 ⋯ p k , {\displaystyle cp_{1}\cdots p_{k},} where c is a real number and each p i {\displaystyle p_{i}} is a monic ...
It may happen that this makes the coefficient 0. [12] Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, [d] a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. A real polynomial is a polynomial with real coefficients.
is a formal product of these indeterminates, possibly raised to a nonnegative power. As usual, exponents equal to one and factors with a zero exponent can be omitted. In particular, = The tuple of exponents α = (α 1, …, α n) is called the multidegree or exponent vector of the monomial. For a less cumbersome notation, the abbreviation
Divide the previously dropped/summed number by the leading coefficient of the divisor and place it on the row below (this doesn't need to be done if the leading coefficient is 1). In this case q 3 = a 7 b 4 {\displaystyle q_{3}={\dfrac {a_{7}}{b_{4}}}} , where the index 3 = 7 − 4 {\displaystyle 3=7-4} has been chosen by subtracting the index ...