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mathematics
a first course in abstract algebra

Questions and Answers of A First Course In Abstract Algebra

Show that a finite field of pn elements has exactly one subfield of pm elements for each divisor m of n.
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2 + √3) over Q(√2 + √3)
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.The vectors in a subset S of a vector space V
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).α = √π, F = Q(π)
Show that xPn - x is the product of all monic irreducible polynomials in Zp[x] of a degreed dividing n.
Find the degree and a basis for the given field extension. Be prepared to justify your answers.Q(√2, √6 + √10) over Q(√3 +√5)
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.The dimension over F of a finite-dimensional
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A basis for a vector space V over a field F is
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).α = π2, F = Q
Let p be an odd prime. a. Show that for a ∈ Z, where a ≠ 0 (mod p), the congruence x2 = a (mod p) has a solution in Z if and only if a(P-1)/2 = 1 (mod p).b. Using part (a), determine whether
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.An algebraic extension of a field F is a field
Mark each of the following true or false. ___ a. The sum of two vectors is a vector. ___ b. The sum of two scalars is a vector. ___ c. The product of two scalars is a scalar. ___
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).α = π2, F = Q(π)
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A finite extension field of a field F is one
Let V be a vector space over a field F. a. Define a subspace of the vector space V over F. b. Prove that an intersection of subspaces of V is again a subspace of V over F.
Classify the given a ∈ C as algebraic or transcendental over the given field F. If α is algebraic over F, find deg(α, F).α = π2 , F = Q(π3)
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.The algebraic closure F̅E of a field F in an
Let V be a vector space over a field F, and let S ={αi |i ∈ I} be a nonempty collection of vectors in V. a. Using Exercise 16(b), define the subspace of V generated by S. b. Prove that
Refer to Example 29.19 of the text. The polynomial x2 + x + 1 has a zero a in Z2(α) and thus must factor into a product of linear factors in (Z2(α))[x]. Find this factorization.Data from in Example
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A field F is algebraically closed if and only
Let Vi, ··· , Vn be vector spaces over the same field F. Define the direct sum V1 ⊕ ··· ⊕ Vn of the vectors spaces Vi for i = 1, ···, n, and show that the direct sum is again a vector
a. Show that the polynomial x2 + 1 is irreducible in Z3[x]. b. Let a be a zero of x2 + 1 in an extension field of Z3 As in Example 29.19, give the multiplication and addition tables for the nine
Show by an example that for a proper extension field E of a field F, the algebraic closure of F in E need not be algebraically closed.
Generalize Example 30.2 to obtain the vector space Fn of ordered n-tuples of elements of F over the field F, for any field F. What is a basis for Fn?Data from in Example 30.2Consider the abelian
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.An element α of an extension field E of a
Mark each of the following true or false. ___ a. If a field E is a finite extension of a field F, then E is a finite field. ___ b. Every finite extension of a field is an algebraic
Define an isomorphism of a vector space V over a field F with a vector space V' over the same field F.
Give a one- or two-sentence synopsis of the proof of Theorem 31.4. Data from Theorem 31.4If E is a finite extension field of a field F, and K is a finite extension field of E, then K is a finite
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.An element β of an extension field E of a
Give a one-sentence synopsis of the proof of Theorem 31.3. Data from Theorem 31.3A finite extension field E of a field F is an algebraic extension of F. Proof We must show that for a ∈ E,
Prove that if V is a finite-dimensional vector space over a field F, then a subset {βi, β2 , ··· , βn} of V is a basis for V over F if and only if every vector in V can be expressed uniquely as
Let F be any field. Consider the "system of m simultaneous linear equations inn unknowns" where aij, bi ∈ F.a. Show that the "system has a solution;' that is, there exist X1, ··· , Xn ∈ F
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A monic polynomial in F[x] is one having all
Let (a+ bi) ∈ C where a, b ∈ R and b ≠ 0. Show that C = R(a + bi).
Let V and V' be vector spaces over the same field F. A function ∅ : V → V' is a linear transformation of V into V' if the following conditions are satisfied for all α, β ∈ V and a ∈ F:a. If
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication.A field E is a simple extension of a subfield
Prove that every finite-dimensional vector space V of dimension n over a field F is isomorphic to the vector space Fn of Exercise 19.Data from Exercise 19Generalize Example 30.2 to obtain the vector
Mark each of the following true or false.____ a. The number π is transcendental over Q.____ b. C is a simple extension of R____ c. Every element of a field F is algebraic over F.____ d. R is an
Prove that x2 - 3 is irreducible over Q(3√2).
We have stated without proof that n and e are transcendental over Q. a. Find a subfield F of R such that π is algebraic of degree 3 over F. b. Find a subfield E of R such that e2 is
Let V and V' be vector spaces over the same field F, and let ∅ : V → V' be a linear transformation. a. To what concept that we have studied for the algebraic structures of groups and rings
What degree field extensions can we obtain by successively adjoining to a field F a square root of an element of F not a square in F, then square root of some nonsquare in this new field, and so on?
a. Show that x3 + x2 + 1 is irreducible over Z2. b. Let a be a zero of x3 + x2 + 1 in an extension field of Z2. Show that x3 + x2 + 1 factors into three linear factors in
Let V be a vector space over a field F, and let S be a subspace of V. Define the quotient space V/S, and show that it is a vector space over F.
Let E be a finite extension field of F. Let D be an integral domain such that F ⊆ D ⊆ E. Show that D is a field.
Let E be an extension field of Z2 and let α ∈ E be algebraic of degree 3 over Z2. Classify the groups (Z2(α), +) and ((Z2(α))*, ·) according to the Fundamental Theorem of finitely generated
Let V and V' be vector spaces over the same field F, and let V be finite dimensional over F. Let dim(V) be the dimension of the vector space V over F. Let ∅ : V → V' be a linear
Prove in detail that Q(√3 +√7) = Q(√3,√7).
Let E be an extension field of a field F and let α ∈ E be algebraic over F. The polynomial irr(α, F) is sometimes referred to as the minimal polynomial for a over F. Why is this designation
Generalizing Exercise 27, show that if √a + √b ≠ 0, then Q(√a + √b) = Q(√a, √b) for all a and b in Q.Data from Exercise 27Prove in detail that Q(√3 +√7) = Q(√3,√7).
Give a two- or three-sentence synopsis of Theorem 29.3. Data from Theorem 29.3Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there exists an extension field E of F and
Let E be a finite extension of a field F, and let p(x) ∈ F[x] be irreducible over F and have degree that is not a divisor of [E : F]. Show that p(x) has no zeros in E.
Let E be an extension field of F, and let α, β ∈ E. Suppose α is transcendental over F but algebraic over F(β). Show that β is algebraic over F(α).
Let E be an extension field of F. Let α ∈ E be algebraic of odd degree over F. Show that α2 is algebraic of odd degree over F, and F(α) = F(α2).
Let E be an extension field of a finite field F, where F has q elements. Let α ∈ E be algebraic over F of degree n. Prove that F(α) has qn elements.
Show that if F, E, and K are fields with F ≤ E ≤ K, then K is algebraic over F if and only if E is algebraic over F, and K is algebraic over E.
Show that is a subfield of R by using the ideas of this section, rather than by a formal verification of the field axioms. {a + b(2) + c(√2)² a, b, c = Q}
a. Show that there exists an irreducible polynomial of degree 3 in Z3[x]. b. Show from part (a) that there exists a finite field of 27 elements.
Let E be an extension field of a field F. Prove that every α ∈ E that is not in the algebraic closure F̅E of F in E is transcendental over F̅E.
Consider the prime field ZP of characteristic p≠ 0. a. Show that, for p ≠ 2, not every element in ZP is a square of an element of Zp.b. Using part (a), show that there exist finite fields of
Let E be an extension field of a field F and let α ∈ E be transcendental over F. Show that every element of F(α) that is not in F is also transcendental over F.
Show that if E is an algebraic extension of a field F and contains all zeros in F̅ of every f(x) ∈ F[x]. then E is an algebraically closed field.
Show that no finite field of odd characteristic is algebraically closed.
Let F be a finite field of characteristic p. Show that every element of F is algebraic over the prime field ZP ≤ F.
Prove that, as asserted in the text, the algebraic closure of Q in C is not a finite extension of Q.
Use Exercises 30 and 36 to show that every finite field is of prime-power order, that is, it has a prime-power number of elements. Data from Exercise 30Let E be an extension field of a finite
Argue that every finite extension field of R is either R itself or is isomorphic to C.
Use Zorn's lemma to show that every proper ideal of a ring R with unity is contained in some maximal ideal.
Let E be an algebraically closed extension field of a field F. Show that the algebraic closure F̅E of F in E is algebraically closed.
Is x3 + 2x + 3 an irreducible polynomial of Z5[x]? Why? Express it as a product of irreducible polynomials of Z5[x].
Let ∅ be the element of End((Z x Z, +)) given in Example 24.2. That example showed that ∅ is a right divisor of 0. Show that ∅ is also a left divisor of 0.Data from Example 24.2Consider the
Let r and s be positive integers such that gcd(r, s) = 1. Use the isomorphism in Example 18.15 to show that form, n ∈ Z, there exists an integer x such that x = m (mod r) and x = n (mod s).Data
a. State and prove the generalization of Example 18.15 for a direct product with n factors. b. Prove the Chinese Remainder Theorem: Let ai, bi ∈ Z+ for i = 1, 2, ···, n and let gcd(bi, bj)
Consider (S, +,•),where S is a set and + and • are binary operations on S such that (S, +) is a group, (S*, •) is a group where S* consists of all elements of S except the additive
Describe all ring homomorphisms of Z x Z into Z x Z.
Write the polynomials in R[x, y, z] in decreasing term order, using the order lex for power products xmynZs where z < y < X.2xy3z5 - 5x2yz3 + 7x2y2z - 3x3
Find all prime ideals and all maximal ideals of Z6.
Let F be the ring of all functions mapping R into R and having derivatives of all orders. Differentiation gives a map ∅ : F → F where δ(f(x)) = f'(x). Is δ a homomorphism? Why? Give the
Write the polynomials in R[x, y, z] in decreasing term order, using the order lex for power products zmynxs where x < y < z.The polynomial in Exercise 4.Data from Exercise 438 - 4xz + 2yz - 8xy
Find all c ∈ Z5 such that Z5[x]/(x2 + x + c) is a field.
Give an example of a ring homomorphism ∅ : R→ R' where R has unity 1 and ∅(1) ≠ 0', but ∅(1) is not unity for R'.
List, in increasing order, the smallest 20 power products in R[x, y, z] for the order deglex with power products xmynzs where z < y < x.
Find all c ∈ Z5 such that Z5[x]/(x2 + cx + 1) is a field.
Give an example to show that a factor ring of an integral domain may have divisors of 0.
Write the polynomials in order of decreasing terms using the order deglex with power products xmynzs where z < y < x.The polynomial in Exercise 4.Data from Exercise 438 - 4xz + 2yz - 8xy + 3yz3
Let R and R' be rings and let N and N' be ideals of R and R', respectively. Let ∅ be a homomorphism or R into R'. Show that ∅ induces a natural homomorphism ∅* : R/N → R'/N' if ∅[N]
Let R[x, y] be ordered by lex. Give an example to show that Pi < Pj does not imply that Pi divides Pj.
Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp x Zp for some prime p.
The sign of an even permutation is + 1 and the sign of an odd permutation is -1. Observe that the map sgnn : Sn → {l, -1} defined by sgnn(σ) = sign of σ is a homomorphism of Sn onto the
Let G be a group, and let g ∈ G. Let ∅g : G→ G be defined by ∅g(x) = gx for x ∈ G. For which g ∈ G is ∅g a homomorphism?
Let a group G be generated by { ai | i ∈ I}, where I is some indexing set and ai ∈ G for all i ∈ I. Let ∅ : G → G' and µ : G → G' be two homomorphisms from G into a group G', such that
Show that any group homomorphism ∅ : G → G' where |G| is a prime must either be the trivial homomorphism or a one-to-one map.
Referring to the group S3 given in Example 8.7, compute the product (0P0 + 1p1 + 0P2 + 0µ1 + 1µ2 + 1µ3)(1p0 + 1P1 + 0P2 + 1µ1 + 0µ2 + 1µ3) in the group algebra Z2S3.Data from in 8.7
F = E = Z7 in Theorem 22.4. Compute for the indicated evaluation homomorphism.∅5[(x3 + 2)(4x2 + 3)(x7 + 3x 2 + 1)]Data from Theorem 22.4(The Evaluation Homomorphisms for Field Theory) Let F be
let (Q((x)) have the ordering described in Example 25.9. List the labels a, b, c, d, e of the given elements in an order corresponding to increasing order of the elements as described by the relation
let G = {e, a, b} be a cyclic group of order 3 with identity element e. Write the element in the group algebra Z5G in the form re + sa + tb for r, s, t ∈ Z5.(3e + 3a + 3b )4
F = E = C in Theorem 22.4. Compute for the indicated evaluation homomorphism.∅i(2x3 - x2 + 3x + 2)Data from Theorem 22.4Let F be a subfield of a field E. let a be any element of E, and let x be an
let (Q((x)) have the ordering described in Example 25.9. List the labels a, b, c, d, e of the given elements in an order corresponding to increasing order of the elements as described by the relation
Let (Q((x)) have the ordering described in Example 25.9. List the labels a, b, c, d, e of the given elements in an order corresponding to increasing order of the elements as described by the relation
The polynomial 2x3 + 3x2 - 7x - 5 can be factored into linear factors in Z11[x ]. Find this factorization.

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