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Chapter 3 Field Fundamentals

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Book Id: WPLBN0000673788
Format Type: PDF eBook
File Size: 134.86 KB
Reproduction Date: 2005

Title: Chapter 3 Field Fundamentals  
Language: English
Subject: Science., Mathematics, Logic
Publication Date:


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Chapter 3 Field Fundamentals. (n.d.). Chapter 3 Field Fundamentals. Retrieved from

Mathematics document containing theorems and formulas.

Excerpt: Field Extensions. If F is a field and F[X] is the set of all polynomials over F, that is, polynomials with coefficients in F, we know that F[X] is a Euclidean domain, and therefore a principal ideal domain and a unique factorization domain (see Sections 2.6 and 2.7). Thus any nonzero polynomial f in F[X] can be factored uniquely as a product of irreducible polynomials. Any root of f must be a root of one of the irreducible factors, but at this point we have no concrete information about the existence of roots and how they might be found. For example, X2 + 1 has no real roots, but if we consider the larger field of complex numbers, we get two roots, +i and ?i. It appears that the process of passing to a larger field may help produce roots, and this turns out to be correct.


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