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Unformatted text preview: Mathematics 25(b) Spring 2001 The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra was first proved by Gauss (at a very young age!). There are now many proofs of this theorem known. Here we give one which uses a number of things we’ve done in Math 25. Let F be a field, and let F [ x ] denote the set of all polynomials in one variable with coefficients in F . As usual, we denote the field of complex numbers by C . By a root of a polynomial P ( x ) ∈ F [ x ], we mean an element λ ∈ F such that P ( λ ) = 0. According to the Factor theorem (see Curtis, section 20), P ( λ ) = 0 iff x- λ divides P ( x ). Theorem 1: (Fundamental Theorem of Algebra) If P ( z ) ∈ C [ z ] has degree n ≥ 1, then there exists a nonzero a ∈ C and elements λ 1 ,...,λ n ∈ C such that P ( z ) = a ( z- λ 1 ) ··· ( z- λ n ) . To prove the Fundamental Theorem of Algebra (henceforth called the FTA), it suffices to prove the seemingly weaker statement: Theorem 1 : Every nonconstant polynomial...
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