Prove that there are domains R containing a pair of elements
havig no gcd.(Hint: let R...
Prove that there are domains R containing a pair of elements
havig no gcd.(Hint: let R be a subring of F[x], for a field F,
where R consistes of all polynomials with no linear terms, then
show that x5 and x6 have no gcd) .
Prove the theorem in the lecture:Euclidean Domains and UFD's
Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is a
maximal ideal in F[x] if and only if p(x) is irreducible over
F.
Let
D and D'be integral domains.
Let c = charD and c'= charD'
(a) Prove that the direct product D ×D'has unity.
(b) Let a ∈D and b∈D'.
Prove that (a, b) is a unit in D ×D'⇐⇒
a is a unit in D, b is a unit in D'.
(c) Prove that D×D'is never an integral domain.
(d) Prove that if c, c'> 0,
then char(D ×D') = lcm(c, c')
(e) Prove that if c = 0, then char(D...
Let S ⊆ R and let G be an arbitrary isometry of R . Prove that
the symmetry group of G(S) is isomorphic to the symmetry group of
S. Hint: If F is a symmetry of S, what is the corresponding
symmetry of G(S)?
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then
a∣c.
(B) Let p ≥ 2. Prove that if 2p−1 is prime, then p
must also be prime.
(Abstract Algebra)
Let f : R → R be a function.
(a) Prove that f is continuous on R if and only if, for every
open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is
open.
(b) Use part (a) to prove that if f is continuous on R, its zero
set Z(f) = {x ∈ R : f(x) = 0} is closed.
Let X be a set containing infinitely many elements, and let d be
a metrio on X. Prove that X contains an open set U such that U and
its complement Uc = X\U are both infinite
Let R be the real line with the Euclidean topology.
(a) Prove that R has a countable base for its topology.
(b) Prove that every open cover of R has a countable
subcover.