we have the question that 'if F is a field with char 0, prove
that prime subfield of F is isomorphic to the field of Q'. I
already figure out the answer. BUT from the question I have other
question have risen in my brain. 1. what are the official
definition of the kernel of a map and the characteristics of a
field? 2. what is the link between the kernel and the char? 3. are
they equivalent in some...
Use
the definition f '(x) = lim tends to 0 (f(x+h) - f(x) / h) to find
a. The derivative of f(x) = x + (30/x)
b. The derivative of f(x) = 3x^2- 5x + 30
Prove the following:
(a) Let A be a ring and B be a field. Let f : A → B be a
surjective homomorphism from A to B. Then ker(f) is a maximal
ideal.
(b) If A/J is a field, then J is a maximal ideal.
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.
2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all
xin an interval ( a , b ), then f is constant on ( a , b
).
b True or False. The product of two increasing functions is
increasing. Clarify your answer.
c Find the point on the graph of f ( x ) = 4 − x 2 that is
closest to the point ( 0 , 1 ).
(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈
F[x]. Show that the following properties hold:
(a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x).
(b) If g(x)|f(x), then g(x)h(x)|f(x)h(x).
(c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x).
(d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F
\ {0}