Polynomial And Quotient rings T or F

Question 1

a. Q is a field of quotients of Z.

Answer 1

T

Question 2

R is a field of quotients of Z.

Answer 2

F

Question 3

R is a field of quotients of R.

Answer 3

T

Question 4

C is a field of quotients of R.

Answer 4

F

Question 5

If D is a field, then any field of quotients of D is isomorphic to D.

Answer 5

T

Question 6

The fact that D has no divisors of 0 was used strongly several times in the construction of a field F of quotients of the integral domain D.

Answer 6

T

Question 7

Every element of an integral domain D is a unit in a field F of quotients of D.

Answer 7

F

Question 8

Every nonzero element of an integral domain D is a unit in a field F of quotients of D.

Answer 8

T

Question 9

A field of quotients F′ of a subdomain D′ of an integral domain D can be regarded as a subfield of some field of quotients of D.

Answer 9

T

Question 10

Every field of quotients of Z is isomorphic to Q.

Answer 10

T

Question 11

The polynomial (anx^n +···+a1x+a0)∈ R[x] is 0 if and only if
ai =0, fori=0,1,··· ,n.

Answer 11

T

Question 12

If R is a commutative ring, then R[x] is commutative.

Answer 12

T

Question 13

If D is an integral domain, then D[x] is an integral domain.

Answer 13

T

Question 14

If R is a ring containing divisors of 0, then R[x] has divisors of 0.

Answer 14

T

Question 15

If R is a ring and f(x) and g(x) in R[x] are of degrees 3 and 4, respectively, then f(x)g(x) may be of degree 8 in R[x].

Answer 15

F

Question 16

If R is any ring and f(x) and g(x) in R[x] are of degrees 3 and 4, respectively, then f(x)g(x) is always of degree 7.

Answer 16

F

Question 17

If F is a subfield of E and α∈E is a zero of f(x)∈F[x], then α is a zero of h(x)=f(x)g(x) for all g(x) ∈ F[x].

Answer 17

T

Question 18

If F is a field, then the units in F[x] are precisely the units in F.

Answer 18

T

Question 19

If R is a ring with unity, then x is never a divisor of 0 in R[x].

Answer 19

T

Question 20

If R is a ring, then the zero divisors in R[x] are precisely the zero divisors in R.

Answer 20

F