Page 212 - DMTH403_ABSTRACT_ALGEBRA

Unit 20: Principal Ideal Domains

5. Any irreducible polynomials R[x] has degree 1 or define .................. Notes
(a) n (b) 2

(c) 4 (d) 5
20.3 Summary

The definition and examples of a Euclidean domain.

Z any field and any polynomial ring over a field are Euclidean domains.

Units, associates, factors, the g.c.d. of two elements, prime elements and irreducible elements

in an integral domain.
The definition and examples of a principal ideal domain (PID).

Every Euclidean domain is a PID, but the converse is not true.

Thus, Z, F and F[x] are PIDs, for any field F.
The g.c.d. of any two elements a and b in a PID R exists and is of the form ax + by for some

x, y  R.
The Fundamental Theorem of Algebra: Any non-constant polynomial over C has all its

roots in C.
20.4 Keywords

Euclidean Domain: An integral domain D is called a Euclidean domain if for each non-zero
element x in D there is assigned a non-negative integer (x) such that
(i) (ab) (b) for all non-zero a,b in D, and
(ii) for any non-zero elements a,b in D there exist q,r in D such that a = bq + r, where either
r = 0 or (r) < (b).
UID: Let R be a commutative ring with identity. A non-zero element p of R is said to be
irreducible if
(i) p is not a unit of R, and
(ii) if p = ab for a,b in R, then either a or b is a unit of R.
Any principal ideal domain is a unique factorization domain.

20.5 Review Questions

1. Show that a subring of a PID need not be PID.
2. Will any quotient ring of a PID be a PID? Why? Remember that a PID must be an integral
domain.

3. Let R be an integral domain. Show that
(a) u is a unit in R iff u | 1.
(b) for a, b  R, a | b and b | a iff a and b are associates in R.

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