Wednesday, November 18, 2009

Bertrand's Postulate (Theorem)

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In today's blog, I will present a proof for Bertrand's Postulate which is really a theorem. The content is taken primarily from an...
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Primorial

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The primorial is the product of primes less than or equal to a number n . Definition: Primorial: n# The primorial n# is defined as follo...

Pascal's Rule

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Theorem: Pascal's Rule C(n,k) + C(n,k-1) = C(n+1,k) where: Proof: (1) We can find a common denominator for C(n,k) and C(n,k-1) so tha...
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Monday, November 16, 2009

Multiplicity of a Prime Factor

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Definition 1: Multiplicity of a prime factor The multiplicity of a prime factor of a given integer is the highest power of that factor tha...
Monday, October 05, 2009

Common roots with an irreducible polynomial

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The content in today's blog is taken straight from Jean-Pierre Tignol's Galois Theory of Algebraic Equations . Theorem: Let f(x) be...
Saturday, October 03, 2009

A polynomial invariant on all but one variable

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The content in today's blog is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations . Lemma: Let g be a polyno...
Thursday, October 01, 2009

Nonzero Polynomials with Distinct Parameters

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The following is taken from Harold M. Edwards in his book Galois Theory . Theorem: Let K be a field. Let x 1 , x 2 , x 3 , .. . be an infin...

Products of Nonzero Polynomials

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The following is taken from Harold M. Edwards in his book Galois Theory . Theorem: The product of nonzero polynomials is a nonzero polynomi...
Tuesday, September 29, 2009

The Discriminant

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The content in today's blog is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations . For a definition of symme...