Page 219 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes  In a UFD (and hence, in a PID) an element is prime iff it is irreducible.
Any two elements in a UFD have a g.c.d.

If R is a UFD. then so is K[x].

21.3 Keyword

Unique Factorisation Domain: We call an integral domain R a unique factorisation domain
(UFD, in short) if every non-zero element of R which is not a unit in R can be uniquely expressed
as a product of a finite number of irreducible elements of R.

21.4 Review Questions

1. Directly prove that F[x] is a UFD, for any field F.

(Hint: Suppose you want to factorise f(x). Then use induction on deg f(x)).
2. Give two different prime factorisations of 10 in Z.
3. Give two different factorisations of 6 as a product of irreducible elements in Z[  5].

4. Give an example of a UFD which is not a PID.
5. If p is an irreducible element of a UFD R, then is it irreducible in every quotient ring of R?
6. Is the quotient ring of a UFD a UFD? Why?

7. Is a subring of a UFD a UFD? Why?

Answers: Self Assessment

1. (b) 2. (b) 3. (a) 4. (c) 5. (c)

21.5 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/
archives.math.utk.edu

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