Theorem 6.29: Any Field is an integral domain ◆[Z;+,×] is an integral domain. But it is not a field
Theorem 6.29: Any Field is an integral domain [Z;+,] is an integral domain. But it is not a field
Theorem 6.30: A finite integral domain is a field integral domain: commutative, no zero-divisor Field: commutative, identity, inverse identity, inverse Let r; t, be a finite integral domain. ◆(1) Need to find e∈ R such that e*a=afor ala∈R ◆(2) For each a∈R-{0}, need to find an element ber such that ab=e Proof: (l)Let r=(,.. an3 ◆Forc∈R,C≠0, consider the set rc={a1tc, a2xC,…,ane∈R
Theorem 6.30: A finite integral domain is a field. integral domain :commutative, no zero-divisor Field: commutative, identity, inverse identity, inverse Let [R;+,*] be a finite integral domain. (1)Need to find eR such that e*a =a for all a R. (2)For each aR-{0}, need to find an element bR such that a*b =e. Proof:(1)Let R={a1 ,a2 ,an }. For cR, c 0, consider the set Rc={a1*c, a2*c, ,an*c}R
Example: Zm+, is a field iff m is a prime number a]=? If GCD(a, n)=1, then there exist k and S, S.t. ak+ns=1, where k,s∈Z ◆ns=1-ak ◆[1l=ak=a]|k ◆|k=|a ◆ Euclidean algorithn
Example: [Zm;+,*] is a field iff m is a prime number [a]-1=? If GCD(a,n)=1,then there exist k and s, s.t. ak+ns=1, where k, sZ. ns=1-ak. [1]=[ak]=[a][k] [k]= [a]-1 Euclidean algorithm
Theorem 6.31(Fermat's Little Theorem): if p is prime number, and gcd(a, p=1, then ap-=l mod p o Corollary 6.3: If p is prime number, aEZ, then ap=a mod p
Theorem 6.31(Fermat’s Little Theorem): if p is prime number, and GCD(a,p)=1, then ap-11 mod p Corollary 6.3: If p is prime number, aZ, then apa mod p
Definition 27: The characteristic of a ring R with 1 is the smallest nonzero number n such that 0=1+1+.,.+1(n times) if such an n exists: otherwise the characteristic is defined to be 0. We denoted by char(r). Theorem 6.32: Let p be the characteristic of a a ring r with 1(e). Then following results hold. ()For VaER, pa=0. And if r is an integral domain, then p is the smallest nonzero number such that 0=la. where a=0. (2)IfR is an integral domain, then the characteristic is either 0 or a prime number
•Definition 27: The characteristic of a ring R with 1 is the smallest nonzero number n such that 0 =1 + 1 + ···+ 1 (n times) if such an n exists; otherwise the characteristic is defined to be 0. We denoted by char(R). Theorem 6.32: Let p be the characteristic of a ring R with 1(e). Then following results hold. (1)For aR, pa=0. And if R is an integral domain, then p is the smallest nonzero number such that 0=la, where a0. (2)If R is an integral domain, then the characteristic is either 0 or a prime number