Basic Abstract Algebra
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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.
Email your librarian or administrator to recommend adding this book to your organisation's collection.