Abstract Algebra Theory and Applications - Computer Science ...

14.2 INTEGRAL DOMAINS AND FIELDS 237are in T , then clearly A − B is also in T . Also,( aa′abAB =′ + bc ′ )0 cc ′is in T .14.2 Integral Domains and FieldsLet us briefly recall some definitions. If R is a ring and r is a nonzero elementin R, then r is said to be a zero divisor if there is some nonzero elements ∈ R such that rs = 0. A commutative ring with identity is said to bean integral domain if it has no zero divisors. If an element a in a ring Rwith identity has a multiplicative inverse, we say that a is a unit. If everynonzero element in a ring R is a unit, then R is called a division ring. Acommutative division ring is called a field.Example 9. If i 2 = −1, then the set Z[i] = {m + ni : m, n ∈ Z} forms aring known as the Gaussian integers. It is easily seen that the Gaussianintegers are a subring of the complex numbers since they are closed underaddition and multiplication. Let α = a+bi be a unit in Z[i]. Then α = a−biis also a unit since if αβ = 1, then αβ = 1. If β = c + di, then1 = αβαβ = (a 2 + b 2 )(c 2 + d 2 ).Therefore, a 2 + b 2 must either be 1 or −1; or, equivalently, a + bi = ±1or a + bi = ±i. Therefore, units of this ring are ±1 and ±i; hence, theGaussian integers are not a field. We will leave it as an exercise to provethat the Gaussian integers are an integral domain.Example 10. The set of matrices{( ) ( 1 0 1 1F =,0 1 1 0with entries in Z 2 forms a field.) ( 0 1,1 1) ( 0 0,0 0)}Example 11. The set Q( √ 2 ) = {a + b √ 2 : a, b ∈ Q} is a field. The inverseof an element a + b √ 2 in Q( √ 2 ) isaa 2 − 2b 2 + −b √2.a 2 − 2b 2We have the following alternative characterization of integral domains.