Irreducibility Tests in Q[T]
by Keith Conrad · University of Connecticut
Expository note on two practical irreducibility tests for monic integer polynomials over the rationals: reduction mod p and Eisenstein's criterion, with Gauss's lemma linking Z[T] to Q[T], shifted-variable tricks, limitations of each test, and many worked examples. Factorization is the core practical skill of the topic and textbooks scatter the tests; this collects them in one rigorous, example-driven place.
More resources on Polynomial Rings
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Ideals, Varieties, and Algorithms (5th Edition)
Undergraduate text on multivariable polynomial rings: monomial orders, the multivariable division algorithm, Groebner bases and Buchberger's algorithm, the Hilbert basis theorem, elimination theory, and the ideal-variety correspondence, with computer algebra exercises. Chapters 1-2 are the definitive accessible treatment of ideals in k[x1,...,xn] and the multivariable division algorithm.
Abstract Algebra: Theory and Applications, Chapter 17: Polynomials
Cleanest free first pass at single-variable polynomial rings. Open-textbook chapter defining polynomial rings over a ring, proving the division algorithm for F[x], and covering irreducible polynomials, Eisenstein's criterion, and ideals in F[x], with exercises and Sage computations.