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Unformatted text preview: 1.8. POLYNOMIALS 49 Proof. The proof is by induction on the degree. Every polynomial of de gree 1 is irreducible, by the definition of irreducibility. So let f be a polynomial of degree greater than 1, and make the inductive hypothesis that every polynomial whose degree is positive but less than the degree of f can be written as a product of irreducible polynomials. If f is not itself irreducible, then it can be written as a product, f D g 1 g 2 , where 1 deg .g i / < deg .f / . By the inductive hypothesis, each g i is a product of irreducible polynomials, and thus so is f . n Proposition 1.8.9. KOExŁ contains infinitely many irreducible polynomials. Proof. If K is an field with infinitely many elements like Q , R or C , then f x k W k 2 K g is already an infinite set of irreducible polynomials. However, there also exist fields with only finitely many elements, as we will see later. For such fields, we can apply the same proof as for Theorem 1.6.6 (replacing prime numbers by irreducible polynomials).(replacing prime numbers by irreducible polynomials)....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Polynomials

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