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Monday, February 20, 2012

Abstract Algebra E-222 video 25 Rings 2

Every word counts in mathematics.

Every non zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. ( Fundamental theorem of algebra, Wikipedia )

The polynomial $x^2-1=0$ has ( thus ) two roots: $(1, -1)$. However, if we consider the coefficients of the polynomial as elements of the ring $\mathbf{Z/8Z}$ then the polynomial has four roots: $(1, -1, 3, -3)$.

$x^2-1$ has $4$ roots...

In lecture 25 professor Gross explains how the division and Euclidean Algorithm can be applied to polynomials in Polynomial Rings over a field.

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)