Algebraic extensions of fields
P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul · Cambridge University Press eBooks · 1994
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, x 2 + 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.