Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- Glossary of symbols
- Part I Preliminaries
- Part II Groups
- Part III Rings and modules
- Part IV Field theory
- Chapter 15 Algebraic extensions of fields
- Chapter 16 Normal and separable extensions
- Chapter 17 Galois theory
- Chapter 18 Applications of Galois theory to classical problems
- Part V Additional topics
- Solutions to odd-numbered problems
- Selected bibliography
- Index
Chapter 15 - Algebraic extensions of fields
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the second edition
- Preface to the first edition
- Glossary of symbols
- Part I Preliminaries
- Part II Groups
- Part III Rings and modules
- Part IV Field theory
- Chapter 15 Algebraic extensions of fields
- Chapter 16 Normal and separable extensions
- Chapter 17 Galois theory
- Chapter 18 Applications of Galois theory to classical problems
- Part V Additional topics
- Solutions to odd-numbered problems
- Selected bibliography
- Index
Summary
Irreducible polynomials and Eisenstein criterion
Let F be a field, and let F[x] be the ring of polynomials in x over F. We know that F[x] is an integral domain with unity and contains F as a proper subring. A polynomial f(x) in F[x] is called irreducible if the degree of f(x) ≥ 1 and, whenever f(x) = g(x)h(x), where g(x),h(x) ∈ F[x], then g(x) ∈ F or h(x) ∈ F. If a polynomial is not irreducible, it is called reducible.
We remark that irreducibility of a polynomial depends on the nature of the field. For example, x2 + 1 is irreducible over R but reducible over C.
Properties of F[x]
We recall some of the basic properties of F[x].
(i) The division algorithm holds in F[x]. This means that if f(x) ∈ F[x] and 0 ≠ g(x) ∈ F[x], then there exist unique q(x), r(x) ∈ F[x] such that f(x) = g(x)q(x) = r(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).
(ii) F[x] is a PID (Theorem 3.2, Chapter 11).
(iii) F[x] is a UFD (Example 1.2(b), Chapter 11).
(iv) The units of F[x] are the nonzero elements of F.
(v) If p(x) is irreducible in F[x], then F[x]/(p(x)) is a field, and conversely.
We now proceed to prove some results for testing whether a polynomial is reducible or irreducible.
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- Information
- Basic Abstract Algebra , pp. 281 - 299Publisher: Cambridge University PressPrint publication year: 1994