If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0. a) Prove that...

If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and h(?) ≠ 0.

a) Prove that [?(?)] = [?(?)] if and only if ?(?) ≡ ?(?)(???( (h(?)).

b) Prove that congruence classes modulo h(?) are either disjoint or identical.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

we have the question that 'if F is a field with char 0, prove that prime...

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...

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R)....

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R). Explain why {f(x) : f(0) = 1} is not.

Prove: Every root field over F is the root field of some irreducible polynomial over F....

Prove: Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)

prove that a ring R is a field if and only if (R-{0}, .) is an...

prove that a ring R is a field if and only if (R-{0}, .) is an abelian group

Use the definition f '(x) = lim tends to 0 (f(x+h) - f(x) / h) to...

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

Let F be a finite field. Prove that the multiplicative group F*,x) is cyclic.

Prove the following: (a) Let A be a ring and B be a field. Let f...

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...

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...

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...

(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}

Subjects