Download Multiple choice questions on Ring and Field Theory and more Quizzes Abstract Algebra in PDF only on Docsity!
Let Rand S be any two rings. Suppose that :R’s be a ring homomorphism if
(1) o(a+b) = (a) + o(b)
(2) o(a.b) = (a). ¢(b) (3) (1) and^ (2)^ both^ hold (4) None
Which of the following isa ring homo
morphism. (1) :Rx]’R given by
(2) :R[X]’R[X] given by
[f(X)]=f(1) Vf(x)eR[X]
o[f(«)]=2.f(x) vf(x)eR[X]
(3) 0:R[X]’R[X] given by [f(x)]=1 vf(x)eR[X] (4) None The quotient field of Z,the ring of integers is (1) R (2) Q (3) C
Let : RS be a ring homomorphism
from a ring R into a ring S. Then Ker is (1) Prime ideal (2) Maximal ideal (3) Subring but not ideal (4) None
Let o be a ring homomorphism from a
ring Rintoa ring S. Then is one-one (injective) if and only if - (1) Ker o{0} (2) Ker ¢={0} (3) Ker o is an ideal (4) None Let Ibe an ideal of a ring R. Then the map :R’R/I given by (x)=x+I is
onto
.orphism but not
(2) An isomorphism (3) An onto ring homomorphism (4) None
Let R be an integral domain with uni
ty. Then polynomial ring R[x] is
(1) An integral domain
(2) Commutative ring without unity
(3) Non-commutative ring (4) None
Let p be aprime number. Then pZ
is a (1) Prime ideal (2) Maximal ideal
(3) Both (1) and (2)
(4) None
Z /5Z is (1) Non-commutative ring (2) Field
(3) Commutative ring without unity
10. Let D=Z [x]. Then quotient field of
D is (1) D (2) Q
(3) f(x) e D
(4) None
- Let F be afield then F[x] is (1) Field (2) Non-commutative ring (3) Commutative ring without unity (4) None
12. Let Fbe a field and f(x) eF[x], g(x)
eF[X] with g(x)=0, then there exists
unique r(x), q(x) in F[x] such that
f(x)=q(x).g(x) +r(x) where either
r(x)=0 or deg r(x)<deg g(x). This
theorem is knówn as (1) Division algorithm (2) Factor theorem
(3) Gauss lemma
(4) None
- Let (^) f(x)=x3+2x+2 and (^) g(x)=2x+ where f(X), g(x)e Z, [X]. Then
f(x).g(x) is.
(1) 2x+2x2+x+ (2) 2x3+x2+2x+
(3) 2x4+x3+x²+x+
(4) None
- Let F be a field and let f(x)=
a,xta,xn-l+..... +a, in F[x]. Then
X-1 is a factor of f(x) if and only if
(1) a,ta,-t..
(2) a,. a-1... à,=
..ta,a, =
(3) a,ta,t .....ta,=
(4) None
21. The polynomial 2x2+6 is
(1) Reducible over Q
(2) Reducible over Z (3) Irreducible over Z (4) None
- The polynomial x²+5 is. (1) Reducible over Q (2) Reducible over R
(3) Reducible over C
(4) None
- Let f(x) and g x) be any two primitive polynomials over Z.Then content of
f(x).g(x) over Z is
(3) Greater than 1 (4) None
24. "Let f(x)=a,x"+a,,x-l+... ta,eZ
[x]. If there is a prime p such that
pxa,, p/an-1 ,P/a, and p²/a, then
f(x) is irreducible over Q", This state ment is known as (1) Gauss lemma (2) Division algorithm (3) Factor theorem (4) Eisenstein's criterion
- The polynomial 20x3+10x+20 is (1) Reducible over Q (2) Irreducible over Q (3) Divisible by x+ (4) None
- Let F be a field and p(x) is an
irreducible polynomial over F. Then.
3x5+15x+
(1) <p(x)> is a maximal ideal (2) F[x]/<p(x)> is a-field (3) Option (1) and (2) both are true (4) None
27. LetF be afield and a(x), b(x) eF[X].
If p(x) eF[X] is an irreducible poly
nomial over F such that p(x) |a(x).
b(x), then
(1) P(x)la(x) or p(x)|b(x)
(2) p(x)la(x) but p(x)| b(x) (3) p(x)|b(x) but p(x)la(x)
- The value (X+1+<+1>) is (1) x2+x+<x*+1> (2) x+x²4<x2+1> (3) 1+X+<x2+1> (4) None
of (x+<*+1>).
- 1 + -3¬Z[-3] is (1) Prime
(2) Prinme and irreducible both
(3) Irreducible but not prime (4) None
30. Let F be an integral domain. Then fol
lowing is true:
(1) Every irreducible is prime
(2) Every prime is irreducible (3) p is prime p is irreducible
(4) None
- In z[V5], 2 is (1) Prime (2) Irreducible (3) Prime but not irreducible (4) None
- The quotient ring (1) Non-commutative ring
(2) Commutative ring but not a
field (3) Field (4) None
- The field
phic (1) R (2) C
<1+x²>Q[x]
(3) C(i)
is a
<1+XQLX] +x²> is^ isomor
- The
<1+x'>Z,^ [x]
(2) 9 (3) 4
number
(4) None
(1) R
(2) Q[7]
- The splitting field of x2-7eQ[x] is
(3) Q(V7) (4) None
is
of
- LetE be a splitting field of polynomi al f(x) eF[X], then
elements in
(1) E is asmallest field extension of Fcontaining all zeroes of f(x) (2) E is afield extension which has only one zero of f(x)
(4) None
(3) E is afield extension of F having no zeroes of f(x)
43. Let^ p(x)^ be^ an^ irreducible^ polynomial
over afield F. If a is any zero of f(x) in some extension field E of F and b is a zero of f(x) in some extension E of F then (1) F(a)=F(b) (2) F[a]=-F[b] (3) F(a)F(b) (4) None
44. Splitting field of x³-1 eQ[x] is
(1) Q)
(2) Q(-)
(3) Both (1) and (2)
(4) None
- Which of the following is true? (1) [R:Q]=
(2) [Q(r):Q]=
(3) Both (1) and (2)
(4) None
- Which^ one^ of^ the^ following^ is^ true? (1) Q/2)^ is^ an^ algebraic^ extension of Q
(2) n is^ not^ algebraic over Q
(3) is^ algebraic^ over^ R
(4) All of the above
- The^ minimal^ polynomial^ of^5 over Q is -
(1) x-/
(2) x²- (3) (x2-5)² (4) None
- Which of the following is true? (1) [Q/3): Q] = 3 (2) (Q/3):Q] +[Q(/5)}:Q| (3) Q(/3,/5)= Q/3 +/5) (4) None
- The order |G(QV2)/Q)|of Galois group G(Q/2)/Q) is (1) 3 (2) 2 (3) 1
- Let L be a finite field extension of K. Then (1) |G(UK)I<[L:K]
(2) |G(L/K)|2[L:K]
(3) |G(L/K)|>[L:K]
(4) None
57. Which one of the following is true?
(1) |G(Q7)/Q)|s 2 (2) |G(QV7)/)| (3) |G(Q/7)/)|< (4) None
- Splitting field of x?-4eQ[x] is (1) R (2) Q (3) Z (4) None
- Splitting field of x²+1 eQ[x] is (1) Q (2) Q(w) (3) C (4) None
- Let f(x)eK[Xx]^ be^ a^ polynomial^ of degreenand L is^ asplitting^ field^ of
f(x) over K. Then
(1) [L:K]= n (2) [L:K] > n (3) [L:K]<n (4) None
61. Let K be^ a^ field^ and^ f(x)^ eK[x].^ Let^ L,
and L, be^ splitting^ fields^ of^ f(x)^ over
K. Then
(1) L, and L, both are same (2) L, # L, and L, and L, are not isomorphic (3) L, and L, are unique up to K-isomorphism (4) None
- (^) Which one of (^) the (^) following is (^) true? (1) R (^) is (^) algebraically closed. (2) (^) Every finite (^) field is (^) algebraically closed.
(3) Q1)
- Ris (^) not (^) algebraicaly (^) closed (^) because
(1) x²-1 has all its roots in R
(2) (^) x2+2 (^) does not (^) have root in (^) R
(3) x2-2 has all its roots in R
(4) None
- (^) Which of (^) the (^) following is (^) not (^) true? (1) [Q(/2): Q] + (2) (^) Q(W2) is afinite (^) extension ofQ (3) [QV3):Q]= (4) None
- Let F be a field then (1) Char F=0 always (2) Char F=prime number (3) Char F is either 0 or a prime number (4) None
66. LetK be a finite.field with character
istic 2. Then possible values of |K| is (1) |K]=2^ for some n (2) |K|= (3) |K|=
- An angle e is constructible if and only if (1) sin 0is constructible but cos is
(2) cos 0 is constructible but sin 9 is
not constructible
(3) Both sin e and cos are not
not constructible
(4) None
- Which one of the following number is
constructible
not constructible?
2T 17
4
(4) None
- Q/2, V3) is an n-radical extension of Q. Then value of n is (1) 2 (2) 3
(4) None
