Polynomials and fields · Definition

Unique factorization domain

modern algebrarings

Before you start you need to know

This is the setting where the fundamental theorem of arithmetic holds: every element is built in an essentially unique way from irreducible pieces.

Definition

An integral domain \(D\) in which every nonzero nonunit element \(a\) is a product of irreducibles \(p_i\), and this factorization is unique up to order and associates.

\[a=p_1p_2\cdots p_r\]

The model is \(\mathbb{Z}\): every integer greater than 1 is a product of primes in only one way, up to order. "Unique" means that two factorizations \(a=p_1\cdots p_r=q_1\cdots q_s\) satisfy \(r=s\) and, after reordering, each \(p_i\) is an associate of \(q_i\). Polynomial rings \(F[x]\) over a field are UFDs too, and so is \(\mathbb{Z}[i]\).

In a UFD every irreducible is prime, and uniqueness fails exactly when some irreducible is not. That is why \(\mathbb{Z}[\sqrt{-5}]\), where \(6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})\), is not a UFD. Also, in a UFD greatest common divisors exist: take each irreducible with the smaller exponent that appears, just as with integers.

The hierarchy is: field ⊂ Euclidean ⊂ principal ideal ⊂ UFD ⊂ integral domain, and none of the inclusions is an equality. Gauss's theorem states that if \(D\) is a UFD, so is \(D[x]\); thus \(\mathbb{Z}[x]\) and \(F[x,y]\) are UFDs even though they are not principal ideal domains.

Think about it

Why does \(\mathbb{Z}\) satisfy uniqueness "up to associates" and not exact uniqueness?

Because \(6=2\cdot3=(-2)\cdot(-3)\): the factors change sign, but they are associates of the previous ones. The units \(\pm1\) force us to speak of associates.

If in a UFD an irreducible \(p\) divides \(ab\), what can you say?

That \(p\) divides \(a\) or \(b\): in a UFD every irreducible is prime.

Exercises 10

  1. In \(\mathbb{Z}\), what is the factorization of 60 into positive irreducibles?
  2. What is the factorization of \(x^4-1\) into irreducibles of \(\mathbb{Q}[x]\)?
  3. Using \(360=2^3\cdot3^2\cdot5\) and \(84=2^2\cdot3\cdot7\), what is the greatest common divisor of 360 and 84?
  4. Which of these rings is a unique factorization domain?
  5. Which implication between types of domains is correct?
  6. In \(\mathbb{Z}[\sqrt{-5}]\), the element 6 has the factorizations \(2\cdot3\) and \((1+\sqrt{-5})(1-\sqrt{-5})\). Since the four factors are irreducible and not associates of each other, what follows?
  7. Which ideal of \(\mathbb{Z}[x]\) shows that \(\mathbb{Z}[x]\) is not a principal ideal domain, even though it is a UFD?
  8. In \(\mathbb{Q}[x]\), how many irreducible factors (with multiplicity, up to associates) does \(x^4-x^2\) have?
  9. What is the factorization of \(x^4+4\) into irreducibles of \(\mathbb{Q}[x]\)?
  10. In \(\mathbb{Z}\), how many positive divisors does 360 have, using its factorization \(2^3\cdot3^2\cdot5\)?

References

  • David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd edition, Wiley, 2004. §8.3
  • Joseph A. Gallian, Contemporary Abstract Algebra, 9th edition, Cengage, 2017. cap. 18
  • I. N. Herstein, Abstract Algebra, 3rd edition, Wiley, 1996. §3.7
  • John B. Fraleigh, A First Course in Abstract Algebra, 7th edition, Pearson, 2002. §45
  • Serge Lang, Algebra, Revised 3rd edition, Springer, 2002. cap. II §5