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 context?

Expert Solution

Colby Messinger answered 1 year ago

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.

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

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.

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...

Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.

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 ).

1.) Answer the following programs a.) Let f be the following function: int f(char *s, char...

1.) Answer the following programs a.) Let f be the following function: int f(char *s, char *t) { char *p1, *p2; for(p1 = s, p2 = t; *p1 != '\0'&& *p2 != '\0'; p1++, p2++){ if (*p1 == *p2) break; } return p1 -s; } What is the return value of f(“accd”, “dacd”)? b.) What will be the output of the below program? #include <stdio.h> int main() { char x[]="ProgramDesign", y[]="ProgramDesign"; if(x==y){ printf("Strings are...

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