If A = {a, b, c, d, e}, B = {a, c, e, f}, and C = {b, d, f, h}, which operation is equivalent to (A\ B) ∩ C?

	a. A\(B∩C)
	b. (A∩C)\(A∩B)
	c. (A∩B)\(A∩C)
	d. ∅

.

If some set A has 7 elements, how many permutations does A have?

	a. 128
	b. 5040
	c. 49
	d. 1

.

If some set A has 7 elements, how many subsets does A have?

	a. 128
	b. 5040
	c. 49
	d. 1

.

Let A = {a, b, c}, B = {d, e, f}, and C = {h, i, j} and suppose that f(a) = d, f(b) = f, f(c) = e, g(d) = i, g(e) = j, and g(f) = h. Which of the following represents (g°f)(x)?

	a. {h, i, j}
	b. {h, j, i}
	c. {j, i, k}
	d. {i, h, j}

.

Let a, b, d, and m ∈ Z and also let d = gcd(a, b), with m = lcm(a, b). What is dm?

	a. a²
	b. b²
	c. ab
	d. None of the above

.

Let f be one-to-one and onto, and let g be one-to-one. What is the most that can be said about g°f?

	a. f and g are inverses.
	b. The image of f and the image of g are the same.
	c. g°f is onto.
	d. g°f is one-to-one.

.

Let f: A → B and g: B → C be maps. Which of the following statements is true?

	a. f and g are inverses.
	b. The domain of g°f is B.
	c. If Img( g°f) is C, then f and g are both onto.
	d. A = C

.

Let f: Z → Q be given by f(n) = n/1. What is true about f?

	a. f is onto.
	b. f is one-to-one.
	c. f is not defined.
	d. f is a bijection.

.

Suppose a, b ∈ Z and gcd(a, b) = 1. What is gcd(a²,b²)?

	a. a
	b. b
	c. c
	d. 1

.

Suppose a, b, and c ∈ Z and that gcd(a, b) = 1. If a | bc, then what must be true?

	a. gcd(a, c) = 1.
	b. gcd(a, c) ≠ 1.
	c. gcd (b, c) = 1.
	d. gcd(b, c) ≠ 1.

.

Suppose f: Q → Z given by f(p/q) = p. What is the most that can be said about f?

	a. f is onto.
	b. f is one-to-one.
	c. f is well-defined.
	d. f has an inverse.

.

Suppose we define a relation ~ on R² by stating that (a,b)~(c,d), if and only if a² + b² ≤ c² + d². Which of the following statements is true?

	a. ~ defines an equivalence relation.
	b. ~ is symmetric and reflexive, but not transitive.
	c. ~ is symmetric and transitive, but not reflexive.
	d. ~ is a function.

.

Supposed there is some set A such that aⁿ ∈ A for every a ∈ A, n ∈ Z. Which of the following is not true in all cases?

	a. ak + am must be in A.
	b. (ak)(am) must be in A.
	c. If A is finite, there must be some j in Z for which aj = 1.
	d. All the above

.

What is gcd(57, 95)?

	a. 19
	b. 5
	c. 15
	d. 3

.

What is the largest possible n for 48 ≡ 56 (mod n)?

	a. 1
	b. 4
	c. 6
	d. 8

.

What is the range of the function f(x) = 3x² -1, if f’s domain is {-2, -1, 0, 1, 2}?

	a. R
	b. {-2, -1, 0, 1, 2}
	c. {11, 2, -1, 2, 11}
	d. {35, 2, -1, 2, 35}

.

Which of the following functions is 1-1 and onto from Q to Q?

	a. f(x) = 1/x²
	b. f(x) = 3x – 7
	c. f(x) = ex
	d. f(x) = √2x-1

.

Which of the following is a bijection from R to R?

	a. f(x) = 1/x
	b. f(x) = 2x² – 1
	c. f(x) = ex
	d. f(x) = -2x³

.

Which of the following is a necessary condition for a function f to be invertible?

	a. f has to be onto.
	b. f has to be one-to-one.
	c. f has to be both one-to-one and onto.
	d. f has to be a function.

.

Which of the following is a permutation of A = {1, 2, 3, 4, 5}?

	a. {1, 2, 3, 4, 5}
	b. {2, 4, 5, 1, 3}
	c. {2, 4, 1, 3, 5}
	d. All the above are permutations of A.

.

Which of the following is true about f: Z → Q?

	a. f cannot be one-to-one.
	b. f cannot be onto.
	c. image of f = Q.
	d. f can be a bijection.

.

Which of the following pairs of functions are inverses for all x ∈ R\{0}?

	a. f(x) = 2x – 3, g(x) = 3x + 2
	b. f(x) = 1/x, g(x) = x
	c. f(x) = 1/x, g(x) = 1/x
	d. f(x) = x, g(x) = -x

.

Which of the following represents a relation but not a function?

	a. {(2, 1), (2, -4), (1, 5), (1, 7), (3, 6)}
	b. {(2, 1), (1, -4), (3, 1), (2, 7), (4, 7)}
	c. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}
	d. All of the above are functions.

.

Which of the following represents an equivalence relation?

	a. a ≡ b (mod n)
	b. set of all left cosets of the subgroup H of a set G
	c. set of all (a, b), (c, d) in R² for which a² + b² = c² + d²
	d. All of the above result in equivalence relations.

.

If A = {a, b, c, d, e}, B = {a, c, e, f}, and C = {b, d, f, h}, what is the result of the operation (A\ B) ∩ C?

	a. {f, h}
	b. {b, d}
	c. {a, c, e}
	d. ∅

.

All of the subgroups of Z under ordinary addition must take which form?

	a. <n>
	b. np, where p is prime
	c. nZ, where n ∊ {0, 1, 2, 3, 4, …}
	d. Of singleton integers, since no subgroup of Z greater than order 1 can exist.

.

Fill in the blank. If a group G contains a proper, nontrivial center, then we know G is __________________.

	a. Not abelian
	b. Cyclic
	c. A permutation group
	d. Isomorphic to a normal group of the same order

.

For some cyclic group G suppose a^m = e and aⁿ = e, for some n ≠ m. What must be true?

	a. G must be {e}.
	b. G is abelian.
	c. gcd(m,n) = 1.
	d. mn has to divide the order of G.

.

If a subgroup H has order 9 and H has 5 right cosets in G, what is the order of G?

	a. 4
	b. 14
	c. 15
	d. 45

.

If ϕ: G → H is a group homomorphism and G is abelian, which of the following statements has to be true?

	a. H is abelian.
	b. H has a normal subgroup.
	c. The center of H contains only e.
	d. None of the above

.

If ϕ: G → H is a group homomorphism and G is cyclic, which of the following statements has to be true?

	a. H is abelian.
	b. The center of H contains only e.
	c. H has a cyclic subgroup.
	d. None of the above

.

If ϕ: G → H is a group homomorphism and ϕ is one-to-one, which of the following statements has to be true?

	a. If the range of G = H, then ϕ is an isomorphism.
	b. H is abelian.
	c. H is a subgroup of G.
	d. The range of G in H is {e}.

.

If ϕ: G → H is a group isomorphism, which of the following statements is true?

	a. H is abelian.
	b. H has a normal subgroup.
	c. H is a cyclic group.
	d. ϕ(x) = e in H implies that x is e in G.

.

Let Aut(G) be the set of all isomorphisms from G onto itself. Which of the following statements is true?

	a. Aut(G) is a subgroup of the largest cyclic subgroup in G.
	b. Aut(G) is a subgroup of the symmetric group of set G.
	c. Aut(G) is abelian.
	d. Aut(G) has a prime order.

.

Let G be a group of order 12 and H be a subgroup of G of order 4. How many left cosets of H are in G?

	a. 12
	b. 6
	c. 3
	d. 2

.

Let G be a group of order p². Which of the following statements must be true?

	a. G is abelian.
	b. G has no normal subgroups.
	c. G must have a normal subgroup of order p.
	d. G is isomorphic to an abelian group of order p².

.

Let G be a group, and let H, J be subgroups of G. Which of the following statements is true?

	a. H ∩ J is a subgroup of G.
	b. H + J = G.
	c. H has no trivial subgroups
	d. J only has a trivial subgroup if H has one.

.

Let H and J be two normal subgroups of G. Which of the following statements is true?

	a. H ∩ J is normal in G.
	b. H ∩ J = {e}
	c. H ∩ J = ∅
	d. None of the above

.

Let the center of G, Z(G) = {x ∊ G: xg = gx for all g ∊ G}. Which of the following statements is true?

	a. G is abelian.
	b. Z(G) is isomorphic to a normal group of the same order.
	c. Z(G) is the only proper subgroup of G.
	d. Z(G) = {e}.

.

Suppose a, b ∈ Z and a ≠ b. Consider the cyclic groups <a> and <b>. What is true about <a> ∩ <b>?

	a. <a> ∪ <b> is a permutation group.
	b. <a> ∩ <b> = {e}.
	c. <a> ∩ <b> cannot be trivial.
	d. <a> ∩ <b> has a prime order.

.

Suppose G = {a, b, c, d, e, f, g, h, i, j} is a group under some operation *. What is the order of the largest possible proper subgroup of G?

	a. 1
	b. 5
	c. 2
	d. G cannot contain a proper subgroup.

.

Suppose G has order pⁿ, where p is prime. Which of the following statements has to be true?

	a. G has to have a normal subgroup of order p.
	b. G cannot contain a normal subgroup.
	c. G is isomorphic to the symmetric group of a set of order n.
	d. G has to be abelian.

.

Suppose G is a cyclic group with order n, and let |k| ≤ n. If we know a^k = e, the identity element of G, then what must be true about k?

	a. a must be e.
	b. k must divide n.
	c. n must divide k.
	d. e² cannot be e.

.

Suppose G is a group, and we know for all g, h ∊ G, (gh)ⁿ = gⁿhⁿ . Then, which of following must be true?

	a. G is a symmetric group
	b. G is a permutation group.
	c. G has only trivial subgroups.
	d. G is abelian.

.

Suppose that G were a finite group of order 20 with subgroups H, order 10 and K, order 5. If K is also a proper subgroup of H, which of the following statements is true?

	a. [G:H] = [G:K][H:K]
	b. [G:K] = [G:H][H:K].
	c. [G:H] = [H:K]/[G:K]
	d. [G:K] = [G:H]/[K:K]

.

Suppose that we had a set J = {(x,y)| x, y ∈Z}. What is true about <J,⊕>, where ⊕is defined as (a,b) ⊕(c,d) = (a-c, b-d)?

	a. J under the operation is a group.
	b. The operation is commutative.
	c. The operation is not associative.
	d. The operation splits J into equivalence classes.

.

The set F of all functions with the operator “+” is a group. Which of the following statements is true?

	a. F is not associative under “+.”
	b. F is not commutative under “+.”
	c. F is a non-abelian group under “+.”
	d. F has no nontrivial normal subgroup.

.

Which of the following is a group homomorphism?

	a. G = Z under ordinary addition, M = <2>, f: G → M, f(x) = 2x.
	b. G = 2x2 matrices under cross multiplication, M = 2x2 matrices under matrix addition, f: G → M, f(x) = 3x
	c. G = {1}, M = Z, f: G → M, f(x) = 1/x.
	d. G = S4, M = Z, f:G → M, f(x) = 3/(x – 1).

.

Which of the following is NOT a group?

	a. The set of nXn matrices under matrix addition
	b. The set of integers under ordinary addition
	c. The set of integers under ordinary division
	d. The set of nXn matrices under cross multiplication

.

Which of the following is a group?

	a. The set of nXn matrices under matrix addition
	b. The set of integers under ordinary division
	c. The set of negative numbers under multiplication
	d. None of the above

.

Fill in the blank. A commutative division ring is also called a(n) ______________.

	a. UFD
	b. Integral domain
	c. Ring with unity
	d. Field

.

Fill in the blank. A ring with an identity element under multiplication, such that the additive and multiplicative identities are not the same is called a(n) _______________.

	a. Ring with unity
	b. Commutative ring
	c. Associative ring
	d. Ideal

.

Fill in the blank. Let R be a Boolean ring. Then, for every a in R, a² = a. Then, R must be ______________.

	a. Finite
	b. A non-trivial ideal
	c. Commutative
	d. A field

.

Fill in the blank. Suppose R is a commutative ring such that for a, b, and c in R, ab = ac implies b = c. At the very least, R has to be a(n) ________________.

	a. UFD
	b. Field
	c. Prime ideal
	d. Integral domain

.

Fill in the blank. Suppose R is a ring and I is a subring. If for every a in I and r in R, ar and ra were in I, I would be called a(n) ________________.

	a. Ideal
	b. Principle ideal
	c. Equal to {0} only
	d. UFD

.

Fill in the blank. Supposed R is a commutative ring and I is some ideal in R. I is a prime ideal if _________________.

	a. For every a, b in R, ab = 0
	b. For every a, b in R, if ab is in I, then either a is in I or b is in I
	c. For every a, b in R, ab = 1
	d. For every a, b in R, if ab is in I, then both a and b must be in I

.

Ideals in rings are similar to what structures in groups?

	a. Isomorphisms
	b. Homomorphisms
	c. Normal subgroups
	d. Non-abelian groups

.

If a ring R of square matrices under matrix addition and cross multiplication is isomorphic to commutative ring, then what must be true?

	a. R must be a UFD.
	b. R must be of finite order.
	c. R must be a commutative ring.
	d. The image of R must be the set of rational numbers.

.

If f is a ring homomorphism from ring R to ring T, what must be true?

	a. f must preserve addition only from R to T.
	b. f must preserve addition and multiplication from R to T.
	c. f must have an inverse from T to R.
	d. R must be commutative.

.

If M is an ideal of R such that no other ideals in R may contain M completely, what is M called?

	a. A quotient ring of R
	b. A prime ideal of R
	c. The maximal ideal of R
	d. The kernel of some homomorphism from R to another ring

.

Let f be a ring isomorphism from ring R to ring T. Which of the following is true?

	a. If R is commutative, then T must be commutative.
	b. If x in R is the identity in R, then f(x) must be the identity in T.
	c. The image of R must be T.
	d. All the above

.

Let R be a commutative ring with a as identity and S be a subring with identity b. What must be true?

	a. S must be a maximal ideal.
	b. a = b.
	c. S must be a field.
	d. S = R.

.

Suppose g is a map from ring R to ring T and is onto and one-to-one. What must be true?

	a. g is a ring isomorphism by definition.
	b. g is a ring isomorphism only if g is a ring homomorphism.
	c. T must be commutative if R is commutative.
	d. T = R.

.

Suppose R and S are rings and the maximal ideal in R is R itself. If R is isomorphic to S, then what must be true?

	a. S must be commutative.
	b. The maximal ideal in S must be S.
	c. The kernel of any isomorphism is {0} in R.
	d. All of the above

.

Suppose R is a ring and I is an ideal in R. What is the kernel of a ring homomorphism g: R ==> R/I?

	a. All of R
	b. The maximal ideal in R
	c. I
	d. {0}

.

What is the only difference between a general division ring and a field?

	a. A field has no zero divisors.
	b. A division ring has no unity.
	c. A division ring has no unity.
	d. A division ring may have non-zero elements that have no inverses under multiplication.

.

What is true of an integral domain?

	a. It is always a field.
	b. It is a commutative ring R with unity such that for every a, b in R, if ab = 0, then a or b = 0.
	c. It is a non-commutative ring with unity.
	d. It is a ring R such that for every a in R, with a not equal to zero, a multiplicative inverse exists.

.

What is true of any ring regarding ideals?

	a. Every ring has two trivial ideals.
	b. Every ideal is homomorphic to a non-commutative subgroup.
	c. Every ideal is finite.
	d. Every ring cannot have any ideals unless the ring is commutative.

.

What is true of ring homomorphisms?

	a. They must be onto.
	b. The kernel of the homomorphism must be an ideal in the domain.
	c. The range of the homomorphism must be commutative.
	d. A ring homomorphism may only be one-to-one to be an isomorphism.

.

Which of the following can be separated into unique factors?

	a. Non-commutative rings
	b. Integral domains
	c. UFDs
	d. Modules

.

Which of the following statements about a ring with a zero divisor is true?

	a. A ring with a zero divisor can never be an integral domain.
	b. A ring with a zero divisor may be an integral domain if it is commutative.
	c. A ring with a zero divisor is a UFD.
	d. A ring with a zero divisor is a prime ideal.

.

Which of the following statements is always true of all rings?

	a. All rings are commutative groups under some defined operation of “addition.”
	b. All rings are commutative under some defined operation of “multiplication.”
	c. Rings only have subsets that are distributive under addition and multiplication.
	d. No ring is isomorphic to a field of the same order.

.

Which of the following statements is true of a ring?

	a. It must always be commutative under multiplication.
	b. It must always be of finite order.
	c. It may not have an identity element under multiplication.
	d. The additive and multiplicative identities are not the same element.

.

Which of the following statements is true of a subring of a commutative ring?

	a. A subring of a commutative ring must itself be commutative.
	b. A subring of a commutative ring may not be commutative.
	c. A subring of a commutative ring may not be associative under multiplication.
	d. A subring of a commutative ring is always a prime ideal.

.

Which of the following is a commutative ring?

	a. The set of all 3x3 matrices under matrix addition and cross multiplication
	b. The set of integers under ordinary addition and multiplication
	c. The set of integers under ordinary multiplication and division
	d. None of the above

.

An extension field E is algebraic over some field F if which of the following is true?

	a. Every element in E solves polynomials over F.
	b. Every element in E is also in F.
	c. E is C.
	d. None of the above

.

Fill in the blank. If E is a finite extension over F and K is a finite extension over E, then K over F has order _______________.

	a. [K:F][E:F]
	b. [K:E][E:F]
	c. 2[K:E]
	d. 2[K:F]

.

Fill in the blank. If E is an extension field over some field F, then F is called the _________________.

	a. Maximal subfield of E
	b. Base field of E
	c. Main field of E
	d. Indention field of E

.

Fill in the blank. If every element that is algebraic in some field F is contained in E, we say that E is ___________________.

	a. The maximum extension field of F
	b. The algebraic closure of F
	c. C
	d. An infinite extension of F

.

Find the splitting field for x⁴- 10x² +21over Q?

	a. R
	b. C
	c. Q(√3)
	d. Q(√3,√7)

.

If K is an extension field of order 15 over some field F and is an extension field of order 3 over some field E that contains F, then what is the order of E over F?

	a. 15
	b. 5
	c. 3
	d. 1

.

Let E be an finite extension field over F with degree n. Which of the following is true?

	a. E can be thought of as a vector space.
	b. E must be an algebraic closure.
	c. E cannot be an extension of a smaller extension over F.
	d. All of the above

.

Let E be the splitting field for a polynomial p(x) with coefficients in F of degree n. What must be true?

	a. E is a infinite field.
	b. [E:F]≤ n.
	c. [E:F] = n + 1.
	d. None of the above

.

Let f(x) = x²+1over Q. Which of the following values is algebraic over Q for this polynomial?

	a. √2
	b. √3
	c. i
	d. 1 + i

.

Let f(x) = x²+x+1 be a polynomial over Q. Find the smallest extension field F over Q.

	a. F = Q(-1+√2)
	b. F = Q(-1+√3)
	c. F = Q(-1+i√3)
	d. F = Q(-1+i√2)

.

Let θ: F ==> E be a field isomorphism with p(x) being a non-constant polynomial in F(x) and q(x) being the corresponding non-constant polynomial in E(x). What must be true?

	a. If K is a splitting field of p(x) and L is a splitting field of q(x), then θ extends to an isomorphism from K to L.
	b. F and E must have the same algebraic closures.
	c. 1 in F must be 0 in E.
	d. All of the above

.

Suppose E = Q(√2+√3 ). What must be contained in E?

	a. All square roots
	b. √6
	c. ∛2
	d. √3-√2

.

Suppose E is an extension field of order 2 over F, and K is an extension field of order 3 over E. Then, K is an extension field of what order over F?

	a. 2
	b. 4
	c. 6
	d. 8

.

Suppose G is an extension field over F with order 5 and F is an extension field over E with order 7. Then, G would be an extension field over E with what order?

	a. 35
	b. 12
	c. 7
	d. 5

.

Suppose we know that a field F is algebraically closed. What has to be true?

	a. There is an extension field E for which any given polynomial in F has a solution.
	b. Every non-constant polynomial in F(x) has a solution in F.
	c. The only extension field of F that contains every solution of F(x) is C.
	d. F cannot be Q.

.

Suppose {1,√2,√3,√6} is the basis of an extension field over Q. What is the smallest such extension field?

	a. Q(√2,√3)
	b. Q(√2,√6)
	c. Q(√3,√6)
	d. R is the smallest such field extension.

.

Supposed K is a splitting field that has order 9 over some field F. What has to be true?

	a. K has to be an algebraic closure of F.
	b. A polynomial in F can be split into 9 unique factors.
	c. There are less than 9 total solutions to any polynomial over F.
	d. F must have order of 3.

.

What is the basis of the field extension Q(√3,√6) over Q?

	a. {√3,√6}
	b. {1, √2, √3,√6}
	c. {1, √3,√6}
	d. {√2}

.

What is the basis of the field extension Q(√3,√7 ) over Q?

	a. {1}
	b. {1,√3,√7}
	c. {√3,√7}
	d. {1,√3,√7,√21}

.

What is the degree of the field extension Q(√3,√6) over Q?

	a. 2
	b. 3
	c. 4
	d. 6

.

Which of the following is a commutative ring, but not a field, under addition and multiplication?

	a. Complex numbers
	b. Real numbers
	c. Rational numbers
	d. Integers

.

Which of the following statements is true?

	a. C cannot be transcendental over Q
	b. All nth roots of integers are algebraic over Q
	c. Extension fields must be finite.
	d. All the above statements are true.

.

Which of the following statements is true?

	a. If F is a finite extension over Q, it must be an algebraic extension of Q.
	b. If F is an infinite extension over Q, it can be a transcendental extension of Q.
	c. C is a transcendental extension over Q.
	d. All of the above statements are true.

.

√2 is algebraic over Q for which of the following polynomials?

	a. f(x)=x²+x+1
	b. f(x)=x²+2x+1
	c. f (x)=x²+2
	d. f(x)=x²-2

.

Fill in the blank. If F is a field and E is a field that contains F, we say that E is an extension of F, _________________.

	a. If F is commutative
	b. If at least one solution to some polynomial over F is in E
	c. If E is C
	d. If C is E

.