Suppose G is a group of order 21, and H is a proper subgroup of order 3. How many distinct left cosets of H are there?

	a. 18
	b. 24
	c. 7
	d. 3

.

Suppose G is a finite group of order 56, and H and K are proper subgroups of G, such that K is also a proper subgroup of H. If [G:K] = 14 and [H:K] = 2, what would [G:H] have to be equal to?

	a. 7
	b. 12
	c. 24
	d. 1

.

Let G be a group of order p⁴, where p is prime. Which of the following statements must be true?

	a. G is cyclic.
	b. G has a nontrivial center.
	c. G is isomorphic to S4
	d. G has no proper subgroups.

.

Let G be a group of order 125. What is true of G’s center?

	a. G’s center must be order 5 or 25.
	b. G has no trivial center.
	c. G is cyclic.
	d. G has no proper subgroups.

.

Suppose G is a finite abelian group. Suppose the center of G has order 11. Then, which if the following must be true?

	a. G cannot be its own center.
	b. G cannot be cyclic.
	c. G must have order 11 or order 121.
	d. None of the above

.

Let G be a finite group with proper subgroups H, J and K. Further, let K ⊂ J ⊂ H. If [G:H] = 5, [H:J] = 3 and [J:K] = 7, what is [G:K]?

	a. 15
	b. 7
	c. 1024
	d. 371

.

Suppose G is a group of order 98. Which of the following statements must be true?

	a. G must contain a proper subgroup H of order 49.
	b. G must be abelian.
	c. G cannot contain a nontrivial center.
	d. G must be cyclic.

.

Suppose G is a group with proper subgroups H and K. Further, suppose K ⊂ H and that K has 12 left cosets in G and K has 3 left cosets in H. How many left cosets in G does H have?

	a. 36
	b. 9
	c. 15
	d. 4

.

Suppose that G is an abelian group of order 270. To which of the following is G isomorphic?

	a. Z18XZ15
	b. Z10XZ9XZ3
	c. Z2XZ27XZ5
	d. Z6XZ9XZ5

.

Suppose G is a finite subgroup of order 39. If H is a proper, nontrivial subgroup, what is the smallest order H might have?

	a. 3
	b. 13
	c. 39
	d. G cannot have a proper, nontrivial subgroup.

.

Suppose G is a finite subgroup of order 39. If H is a proper, nontrivial subgroup, what is the largest order H might have?

	a. 3
	b. 13
	c. 39
	d. G cannot have a proper, nontrivial subgroup.

.

Suppose we know that G is a group and that it has no trivial subgroups. Then, which of the following statements is true?

	a. G has a prime order.
	b. G can only be {e}.
	c. G can only be an infinite group.
	d. G cannot have cosets.

.

G is a finite group with proper, nontrivial subgroups H and J. Which of the following statements must be true?

	a. The order of H ∩ J must divide the order of G.
	b. H ∪ J = G.
	c. H has no nontrivial subgroups.
	d. J only has a trivial subgroup if H has one.

.

Let G be a finite group of order 63. If H and J are proper, nontrivial subgroups of G, H ≠ J, what is the largest order H ∩ J may have?

	a. 21
	b. 7
	c. 3
	d. 1

.

Suppose G is some cyclic group order n. If we know for some a ∈ G that a⁵ = e and a¹¹ = e, then what is the smallest order G may take?

	a. 1
	b. 5
	c. 11
	d. 55

.

Suppose G is an abelian group of order 270. Which of the following statements is false?

	a. G will not have a subgroup of order 15.
	b. G cannot have a subgroup of order 3.
	c. G may not have a subgroup of order 30.
	d. G may have a normal subgroup of order 27.

.

Suppose G is some cyclic group of order 91. Let a ∈ G, such that aⁿ = e. What is the largest value n may take?

	a. 1
	b. 7
	c. 13
	d. 91

.

Let G be a group and H be one of G’s subgroups. If H has order 19 and has 5 left cosets in G, what is the order of G?

	a. 5
	b. 19
	c. 95
	d. 190

.

Suppose K is the result of several intersections of proper subgroups of some finite abelian group G, with G having order 1024. What is true of K?

	a. K has to be {e}.
	b. K has to have an order of 2.
	c. K has to have an order of 2n, where n ≤ 10.
	d. K has to be G.

.

Fill in the blank. Let G be a group G, and let H be a proper subgroup in G. If for all x, y ∈ G, and h ∈ H we know that xh = hy ∈ H implies that x = y, then ________________.

	a. G must be abelian
	b. H must be {e}
	c. H must be G
	d. H must be normal

.

Suppose that we know some group G contains the elements {a, b, c, d, e}, where e is the identity. What is the largest subgroup of G?

	a. {e}
	b. {a, e}
	c. {a, c, e}
	d. G

.

Suppose some subgroup H of G has order 5 and suppose H ≠ G. If G is abelian, what is the smallest order G may have?

	a. 5
	b. 25
	c. 125
	d. 625

.

Suppose G is a finite group and for every a ∈ G, n ∈ Z, aⁿ = e. What has to be true about G?

	a. G has to have a prime order.
	b. G cannot be abelian.
	c. G has to have order of 1.
	d. G has to be empty.

.

Let G have a center of order 7 and let the center have 12 left cosets in G. What must be true about G?

	a. G is not abelian.
	b. G has order 84.
	c. G may have subgroups of order 3 and 4.
	d. All of the above

.

Suppose G = {a,b,c,d,e,f,g,h,i,j} is a group. What are the only possible orders of subgroups of G?

	a. 1
	b. 1,2
	c. 1,2,5
	d. 1,2,5,10

.

Compute (5x² + 3x - 4) + (4x² - x + 9) in Z12.

	a. 20x4 + 7x3 + 42x2 + 23x – 36
	b. 9x2 + 2x + 5
	c. 8x4 + 7x3 + 6x2 + 9x
	d. x2 – 4x – 13

.

Fill in the blank. If a commutative ring R has no zero divisors (that is, if ab = 0, then either a = 0 or b = 0), then R is also called a(n) ________________.

	a. Unique factorization ring
	b. Ideal
	c. Integral domain
	d. Module

.

Let R be an integral domain and let 3, b be elements in R. If 3b = 0, then what might b equal?

	a. b might be equal to any multiple of 3.
	b. b must equal 0.
	c. b cannot be a multiple of 3.
	d. b must be the multiplicative inverse of 3.

.

If f(x) is an irreducible polynomial, with f(x) | p(x)q(x), then which of the following is true?

	a. f(x) is a zero divisor.
	b. f(x) = either p(x) or q(x).
	c. f(x) | p(x) or f(x) | q(x)
	d. f(x) = p(x)q(x).

.

Let F be a field and let a(x) = p(x)q(x) + r(x) have coefficients in F. Then, which of the following statements is true?

	a. r(x) must be the zero polynomial.
	b. deg r(x) must be less than deg q(x).
	c. r(x) must be irreducible.
	d. deg r(x) must be zero.

.

Suppose f(x) is a polynomial with rational coefficients. If f(x) can be expressed as the product of two non-constant polynomials p and q, then what must be true of p and q?

	a. p and q cannot have the same degrees.
	b. p and q MUST have the same degrees.
	c. The degrees of p and q must be smaller than the degree of f.
	d. f must be a monic polynomial.

.

If F is an integral domain, and f(x) is a polynomial with coefficients in F, then which of the following is true of f(x)?

	a. f(x) is irreducible.
	b. f(x) must have a unique factorization.
	c. f(x) is either irreducible or factorable.
	d. f(x) is a constant polynomial.

.

If F is an integral domain and every polynomial with coefficients in F can be factored in distinct ways, then what is true of F?

	a. F is a unique factorization domain.
	b. F must be a field.
	c. F must be a module.
	d. F must be a commutative ring.

.

If a ring R of square matrices under matrix addition and cross multiplication is isomorphic to a 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.

.

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.

.

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

.

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

	a. It can never be an integral domain.
	b. It may be an integral domain if it is commutative.
	c. It is a UFD.
	d. It is a prime ideal.

.

B A field is a commutative division ring.

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

.

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

.

Suppose L is a lattice. Then, which of the following is true?

	a. Each element is divisible by all the elements less than itself.
	b. Each element is equal to all other elements in L.
	c. L has a partial order defined on it.
	d. L has to be a Boolean algebra.

.

Which of the following best describes Boolean algebras?

	a. Boolean algebras are ring-like structures that form incomplete lattices.
	b. Boolean algebras are ring-like structures that have no zero elements.
	c. Boolean algebras are ring-like structures that use both set and logic operations.
	d. Boolean algebras are ring-like structures that have no unique upper and lower bounds.

.

What does the following lattice best represent?

	a. A partial ordering of the unique divisors of 18
	b. A random walk between points 1 and 18
	c. A process map of steps in a program, where numbers represent called subroutines
	d. All of the above

.

Boolean algebras are useful for their ability to do which of the following?

	a. Represent complex circuitry
	b. Assist in finding exploits in a computer network
	c. Represent physical security layouts
	d. Locate trends in a social network

.

Which of the following terms is defined as a complemented, distributed lattice?

	a. A module
	b. A partially ordered set
	c. An invertible matrix
	d. A Boolean algebra

.

If a polynomial of degree n is factorable, then what must be true of any factor?

	a. The degree of the factor must be less than n.
	b. The degree of the factor must be 0.
	c. The degree of the factor must be 1.
	d. The degree of the factor must divide n evenly.

.

Let f be a polynomial with rational coefficients and let it be factorable into two polynomials p and q. If deg p(x) = j and deg q(x) = k, then what must the degree of f be?

	a. deg f(x) = 1.
	b. deg f(x) = 0.
	c. deg f(x) = jk.
	d. There is not enough information given to determine the answer.

.

Fill in the blank. The property that says that if a, b are in some set R and (a,b), (b,a) are both in some relation, P implies that a = b is called the _______________ property.

	a. Transitive
	b. Reflexive
	c. Antisymetric
	d. Symmetric

.

Let X = {1, 2, 3, 4, 6, 12, 24} be the set of divisors of 24, and let M = {2,3,4,6} be a proper subset of X. What is the least upper bound of M?

	a. 6
	b. 12
	c. 24
	d. There is no least upper bound.

.

Fill in the blank. Suppose a lattice L has a largest element I and a smallest element O, and there exists binary operations ⋀ and ⋁ on the elements of L. If for every a in L, there exists an a' in L such that a ⋀ a' = O and a ⋁ a' = I, we say that L is _______________.

	a. Invertible
	b. Commutative
	c. Complemented
	d. Associative

.

Which of the following is an example of Boolean algebra?

	a. Power set of some random set X
	b. Any finite group
	c. A group with a prime order
	d. Any commutative ring

.

Fill in the blank. The _______________ of real numbers, denoted by Rⁿ, form a vector space over R.

	a. n-tuples
	b. n-dimensional arrays
	c. Power set of any n elements
	d. Set of all permutations of any n elements

.

What is a vector space?

	a. A vector space is a non-abelian ring made up of n-tuples.
	b. A vector spaces is an abelian ring made up of n-tuples.
	c. A vector space is a non-abelian group extended by scalar multiplication.
	d. A vector spaces is an abelian group extended by scalar multiplication.

.

Choose the best answer to fill in the blanks. A subset of a vector space is itself a vector space (called a subspace), if it is closed under _______________ multiplication and ______________ addition.

	a. Scalar, scalar
	b. Vector, vector
	c. Scalar, vector
	d. Vector, scalar

.

A set of all linear combinations of some vector set V is called which of the following?

	a. V’s combination class
	b. V’s spanning set
	c. V’s linear representation
	d. V’s rank

.

Fill in the blank. Let S be a set of vectors of some n-dimensional vector space V. If there exist scalars a₁ through a_n, such that a₁v₁ + … + a_nv_n = 0, but not all a’s = 0, then S is said to be linearly _______________.

	a. Independent
	b. Combined
	c. Dependent
	d. Represented

.

If S, a set of vectors of V, spans V, then S is said to be a _______________ of V.

	a. Representation
	b. Projection
	c. Basis
	d. Linear combination

.

Fill in the blank. A mapping of a set V onto itself that changes the spanning set of V is called a change of _______________.

	a. Status
	b. Basis
	c. Identification
	d. Projection

.

An nxn matrix M can only change a basis for an n-dimensional vector space if which of the following is true?

	a. M is nonsingular.
	b. M has a 0 determinant.
	c. M has rank less than n.
	d. M cannot be reduced to I by row operations.

.

What is the product of a 5 X 3 matrix and a 3 X 4 matrix?

	a. A 3 X 3 matrix
	b. A 4 X 5 matrix
	c. A 15 X 12 matrix
	d. A 5 X 4 matrix

.

A homomorphism between two vector spaces can always be represented by which of the following?

	a. A power set
	b. An abelian group
	c. A projection
	d. A matrix

.

Fill in the blank. The rank of a matrix must be _______________ the rank of a map it represents.

	a. Less than
	b. Greater than
	c. Equal to
	d. At least one less than

.

Which of the following can serve as a change of basis map for a n-dimensional space?

	a. A lattice of n partially ordered elements
	b. An invertible nxn matrix
	c. Any nxn matrix
	d. A pair of n-tuples

.

Fill in the blank. The composition of linear maps is a ________________.

	a. Scalar
	b. Single vector
	c. Linear map
	d. Non-reversible projection

.

Fill in the blank. The rank of a linear map is also the dimension of the map’s _______________.

	a. Range space
	b. Null space
	c. Domain
	d. Projection space

.

The dimension of a map’s domain is equal to which of the following?

	a. The dimension of its rank
	b. The dimension of its nullity
	c. The sum of its rank and nullity
	d. None of the above

.

Fill in the blank. The kernel of a linear map from vector space V to vector space W is the inverse image of W’s ______________ vector.

	a. Unity
	b. Zero
	c. Largest
	d. Smallest

.

Fill in the blank. The rank of a linear map is only equal to the dimension of the map’s domain if the nullity of the map is _________________.

	a. Exactly zero
	b. Exactly one
	c. The multiplicative identity vector
	d. The rank of the map’s codomain

.

A linear map that is one-to-one must have which of the following?

	a. An invertible matrix
	b. A singular matrix
	c. A non-trivial nullity
	d. rank less than the domain

.

Which of the following statements is true of a vector homomorphism?

	a. It is always onto and one-to-one.
	b. It is always onto but not necessarily one-to-one.
	c. It is always one-to-one but not necessarily onto.
	d. It is not always one-to-one nor onto.

.

What is a vector homomorphism that is one-to-one and onto called?

	a. A linear map
	b. An isomorphism
	c. A projective map
	d. A change of representation

.

An isomorphism maps a zero vector in one space to which of the following?

	a. A non-trivial kernel in another
	b. The zero vector of another
	c. The multiplicative identity of another
	d. The entire span of another

.

An isomorphism preserves which of the following?

	a. Rank
	b. Identity
	c. Structure
	d. All of the above

.

What is an isomorphism?

	a. An isomorphism is a type of equivalence relation.
	b. An isomorphism is a type of lattice.
	c. An isomorphism is a type of Boolean algebra.
	d. An isomorphism is a type of non-invertible function.

.

What is an isomorphism from V onto itself called?

	a. A self-projection
	b. A reversible map
	c. A power map
	d. An automorphism

.

Fill in the blank. Vector spaces are isomorphic if, and only if, the vector spaces have _________________.

	a. A linear map between them
	b. The same identity
	c. The same dimension
	d. The same kernel

.

Fill in the blank. The set of all field automorphisms is a group under _________________.

	a. Set addition
	b. Composition of functions
	c. Translations of axes
	d. Matrix multiplication

.

What is the set of all automorphisms of some field extension E of a field F that fix F elementwise called?

	a. The power set of E over F
	b. The Galois group of E over F
	c. The super group of E over F
	d. The kernel of E over F

.

Fill in the blank. Complex conjugation (f: a + bi ⇒ a – bi) is a(n) ______________ of the complex numbers.

	a. Automorphism
	b. Bifurcation
	c. Splitting field
	d. Radial transformation

.

If E is a field extension of F and f(x) is a polynomial in F|x|, then any automorphism in G(E/F) defines which of the following?

	a. A permutation of the roots of f(x) that lie in E
	b. A projection of the kernel of F in E
	c. A bijection from all elements in E to all elements in F
	d. None of the above

.

Fill in the blank. Let E be an algebraic extension of a field F. If every irreducible polynomial in F|x| with a root in E has ALL its roots in E, then E is called a _________________ extension of F.

	a. Galois
	b. Projective
	c. Normal
	d. Power

.

E being a finite, normal, separable extension of F is the same as saying which of the following?

	a. F is a splitting field over E of a separable polynomial.
	b. E is a Galois group of F.
	c. F is a Galois group of E.
	d. F = EG for some finite group of aut(E).

.

Which of the following statements about the Galois group of E over F is true?

	a. Galois group of E over F is infinite.
	b. Galois group of E over F is abelian.
	c. Galois group of E over F has order equal to F.
	d. Galois group of E over F has a non-trivial center.

.

What is a root of multiplicity of 1 called?

	a. Simple root
	b. Solitary root
	c. Unitary root
	d. Sole root

.

Suppose E is a field extension of F and further suppose there existed an element a in E such that F(a) = E. What is such an element called?

	a. A simple element
	b. A solitary element
	c. A primitive element
	d. A unitary element

.

Fill in the blank. If every element in E, which is an extension of F, is a root of a separable polynomial in F|x|, then E is called a ________________ extension of F.

	a. Separable
	b. Galois
	c. Simple
	d. Primitive

.

Fill in the blank. A Galois group of E over F is a group of automorphisms of E that _______________ F elementwise (that is, f: Aut(E), such that f(a) = a for all a in F).

	a. Fixes
	b. Transposes
	c. Translates
	d. Rotates

.

What is the set of all automorphisms of an extension field that fix elements of the extended field called?

	a. Commutative ring
	b. Abelian group
	c. Extended group
	d. Vector field

.

Fill in the blank. Suppose E is a finite separable extension of a field F, then there must exist an element m in E, such that _______________.

	a. E = F(m)
	b. F = E(m)
	c. F = E
	d. m = e(m)

.

Fill in the blank. If F is a field and G is a subgroup of Aut(F), then the set {a in F, such that f(a) = a for all f in G} is a ________________ of F.

	a. Galois group
	b. Communtator ring
	c. Subfield
	d. Noncommutative subring

.

Fill in the blank. If f(x) in F|x| of degree n has n distinct roots in F’s splitting field E, then f is ________________.

	a. Fixed
	b. Separable
	c. Simple
	d. Complex

.

Suppose f(x) is irreducible over F. If F has characteristic 0, then which is true of f(x)?

	a. f(x) is separable.
	b. f(x) is a linear transformation.
	c. f(x) is the kernel of F.
	d. f(x) is an element of the automorphism of an extension field of F.

.

Fill in the blank. A function f(x) is separable if it has n distinct roots in F’s ________________ field E.

	a. Extension
	b. Splitting
	c. Fixed
	d. Embedded

.

Which of the following of the automorphisms of F is a subfield of F?

	a. A general extension field
	b. A fixed field
	c. A power set
	d. A symmetric group

.

Fill in the blank. A ______________ of a field F is a field E containing an element x, such that E = F(x).

	a. Separable extension
	b. Prime ideal
	c. Quotient ring
	d. Galois group

.

Fill in the blank. The Galois group of some field E over F has a(n) _______________ center.

	a. Trivial
	b. Non-trivial
	c. Normal
	d. Extended

.

Fill in the blank. A simple root of some polynomial has multiplicity of _______________.

	a. 0
	b. 1
	c. 2
	d. 3

.

Fill in the blank. If a polynomial of degree n in F that has fewer than n roots in an extension field E, then E cannot be _________________.

	a. Simple
	b. Complex
	c. Separable
	d. F’s extension field

.

Fill in the blank. If an irreducible polynomial f has characteristic 0, then f is _______________.

	a. Monic
	b. Separable
	c. Simple
	d. Algebraic

.

Fill in the blank. If some extension field E over F does not contain all the roots of F, then E cannot be ________________.

	a. Separable
	b. Primitive
	c. Solvable
	d. Monic

.

A Galois group is a subgroup of which group of automorphisms?

	a. Permutation
	b. Power
	c. Partially ordered
	d. Separable

.