Prove that every finite integral domain is a field. Give an
example of an integral domain which is not a field.
Please show all steps of the proof. Thank you!!
Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . .
. , xn} be a subset of Z. Prove there exists an i, 1 ≤ i ≤ n such
that xi ≥ xj for all 1 ≤ j ≤ n. Prove that Z is an infinite set.
(Remark: How do you tell if a set is infinite??)
Find an example with an explanation of
Integral domain but not domain with factorization
Domain with factorization but not unique factorization
domains
Domain with factorization but not Noetherian domains
Unique factorization domains but not Noetherian unique
factorization domains
Noetherian Domains but not Noetherian Unique factorization
domains
Noetherian Unique factorization domains but not principal
ideal domain
principal ideal domain but not Euclidean domains
Euclidean domain but not field
(a) Prove that Q(sqareroot 5)={a+b sqareroot 5 ; a,b in Z} is a
subring of Z.
(b) Show that Q(sqareroot 5) is a conmutative ring.
(c) Show that Q(sqareroot 5) has a multiplicative identity.
(d) show that Q(sqareroot 5) is a field.(Hint : you want to
mulitply something by he conjugate.)
(Abstract Algebra)
a.Prove that {12a+ 4b | a, b ∈ Z}={4c |c ∈ Z}.
(b) Prove that{20a+ 16b|a, b ∈ Z}={28m+ 32n|m, n ∈ Z}.
(c) Leta, b ∈ Z−{0}. Prove that{x ∈ Z |ab divides x}⊆{x ∈ Z |a
divides x}.
(d) Prove that{16n|n∈Z}⊆{2n|n ∈ Z}.
L ={x^a y^b z^c | c=a+b}
a) Prove that L is not regular.
b)Prove by giving a context-free grammar that the L is context
free.
c)Give a regular expression of the complement L'.
L ={x^a y^b z^c | c=a+b} a) Prove that L is not regular. b)Prove
by giving a context-free grammar that the L is context free. c)Give
a regular expression of the complement L'.