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?

Choose one answer.

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

Question 2

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

Choose one answer.

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

Question 3

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

Choose one answer.

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

Question 4

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)?

Choose one answer.

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

Question 5

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

Choose one answer.

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

Question 6

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?

Choose one answer.

	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.

Question 7

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

Choose one answer.

	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

Question 8

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

Choose one answer.

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

Question 9

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

Choose one answer.

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

Question 10

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

Choose one answer.

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

Question 11

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

Choose one answer.

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

Question 12

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?

Choose one answer.

	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.

Question 13

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?

Choose one answer.

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

Question 14

What is gcd(57, 95)?

Choose one answer.

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

Question 15

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

Choose one answer.

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

Question 16

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

Choose one answer.

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

Question 17

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

Choose one answer.

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

Question 18

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

Choose one answer.

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

Question 19

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

Choose one answer.

	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.

Question 20

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

Choose one answer.

	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.

Question 21

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

Choose one answer.

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

Question 22

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

Choose one answer.

	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

Question 23

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

Choose one answer.

	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.

Question 24

Which of the following represents an equivalence relation?

Choose one answer.

	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.

Question 25

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?

Choose one answer.

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

Question 26

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

Choose one answer.

	a. <n>
	b. n^p, 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.

Question 27

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

Choose one answer.

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

Question 28

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

Choose one answer.

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

Question 29

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

Choose one answer.

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

Question 30

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

Choose one answer.

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

Question 31

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

Choose one answer.

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

Question 32

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

Choose one answer.

	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}.

Question 33

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

Choose one answer.

	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.

Question 34

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

Choose one answer.

	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.

Question 35

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?

Choose one answer.

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

Question 36

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

Choose one answer.

	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.

Question 37

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

Choose one answer.

	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.

Question 38

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

Choose one answer.

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

Question 39

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

Choose one answer.

	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}.

Question 40

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

Choose one answer.

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

Question 41

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?

Choose one answer.

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

Question 42

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

Choose one answer.

	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.

Question 43

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?

Choose one answer.

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

Question 44

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

Choose one answer.

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

Question 45

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?

Choose one answer.

	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]

Question 46

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)?

Choose one answer.

	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.

Question 47

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

Choose one answer.

	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.

Question 48

Which of the following is a group homomorphism?

Choose one answer.

	a. G = Z under ordinary addition, M = <2>, f: G → M, f(x) = 2^x.
	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).

Question 49

Which of the following is NOT a group?

Choose one answer.

	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

Question 50

Which of the following is a group?

Choose one answer.

	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

Question 51

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

Choose one answer.

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

Question 52

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) _______________.

Choose one answer.

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

Question 53

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

Choose one answer.

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

Question 54

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) ________________.

Choose one answer.

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

Question 55

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) ________________.

Choose one answer.

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

Question 56

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

Choose one answer.

	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

Question 57

Ideals in rings are similar to what structures in groups?

Choose one answer.

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

Question 58

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

Choose one answer.

	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.

Question 59

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

Choose one answer.

	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.

Question 60

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

Choose one answer.

	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

Question 61

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

Choose one answer.

	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

Question 62

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

Choose one answer.

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

Question 63

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

Choose one answer.

	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.

Question 64

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?

Choose one answer.

	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

Question 65

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

Choose one answer.

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

Question 66

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

Choose one answer.

	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.

Question 67

What is true of an integral domain?

Choose one answer.

	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.

Question 68

What is true of any ring regarding ideals?

Choose one answer.

	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.

Question 69

What is true of ring homomorphisms?

Choose one answer.

	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.

Question 70

Which of the following can be separated into unique factors?

Choose one answer.

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

Question 71

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

Choose one answer.

	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.

Question 72

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

Choose one answer.

	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.

Question 73

Which of the following statements is true of a ring?

Choose one answer.

	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.

Question 74

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

Choose one answer.

	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.

Question 75

Which of the following is a commutative ring?

Choose one answer.

	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

Question 76

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

Choose one answer.

	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

Question 77

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 _______________.

Choose one answer.

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

Question 78

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

Choose one answer.

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

Question 79

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

Choose one answer.

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

Question 80

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

Choose one answer.

	a. R
	b. C
	c. Q(√3)
	d. Q(√3,√7)

Question 81

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?

Choose one answer.

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

Question 82

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

Choose one answer.

	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

Question 83

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

Choose one answer.

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

Question 84

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

Choose one answer.

	a. √2
	b. √3
	c. i
	d. 1 + i

Question 85

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

Choose one answer.

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

Question 86

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?

Choose one answer.

	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

Question 87

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

Choose one answer.

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

Question 88

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?

Choose one answer.

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

Question 89

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?

Choose one answer.

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

Question 90

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

Choose one answer.

	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.

Question 91

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

Choose one answer.

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

Question 92

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

Choose one answer.

	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.

Question 93

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

Choose one answer.

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

Question 94

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

Choose one answer.

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

Question 95

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

Choose one answer.

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

Question 96

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

Choose one answer.

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

Question 97

Which of the following statements is true?

Choose one answer.

	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.

Question 98

Which of the following statements is true?

Choose one answer.

	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.

Question 99

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

Choose one answer.

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

Question 100

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, _________________.

Choose one answer.

	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