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Unformatted text preview: Math 461 F Spring 2011 Fundamental Theorem of Algebra Drew Armstrong When we proved the impossibility of the classical construction problems, we were in- terested in the existence of certain roots of polynomials. The flavor of what we did is contained in the following example: Suppose that a cubic polynomial f ( x ) with rational coefficients has 1 + 2 as a root. Applying conjugation in the field extension Q Q [ 2] we conclude that 1- 2 is also a root, and then we can use Descartes Factor Theorem to conclude that f ( x ) = x- (1 + 2) x- (1- 2) g ( x ) = ( x 2- 2 x- 3) g ( x ) , for some polynomial g ( x ) of degree 1 with coefficients in Q [ 2]. However, if we use long division to divide f ( x ) by the polynomial x 2- 2 x- 3 we find that g ( x ) in fact has rational coefficients. That is, g ( x ) = ax + b for some a,b Q , in which case- a/b is a root of g ( x ), and hence f ( x ). We conclude that f ( x ) has a rational root. This argument depends vitally on the fact that f ( x ) is cubic; for higher degrees the proof falls apart. In general it is quite hard to tell whether a given polynomial has a root in a given field. (The exception is the field Q in which we can use the Rational Root Test.) Over time people began to suspect that every polynomial has a root in the com- plex numbers C . The precise statement of this is one of the most famous theorems in mathematics....
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