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mathematics
a first course in abstract algebra

A First Course In Abstract Algebra 7th Edition John Fraleigh - Solutions

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 over a field F are linearly independent over F if and only if the zero vector cannot be expressed as
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 vector space V over a field F is the minimum number of vectors required to span V.
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 a set of vectors in V that span V and are linearly dependent.
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 or not the polynomial x2 - 6 is irreducible in Z17[x]
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 F(α1, α2, ···, αn) where each αi is a zero of some polynomial in F[x].
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. ___ d. The product of a scalar and a vector is a vector.___ e. Every vector space has a finite
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 that can be obtained by adjoining a finite number of elements to F.
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 extension field E of F is the field consisting of all elements of E that are algebraic over F.
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 the vectors in the subspace of V generated by S are precisely the (finite) linear combinations of
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 29.19The polynomial p(x) = x2 + x + 1 in Z2[x] is irreducible over Z2 by Theorem 23.10, since
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 if every polynomial has a zero in F.
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 space over F.
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 elements of Z3(α), written in the order 0, 1, 2, α, 2α, 1 + α, 1 + 2α, 2 + α, and 2 +
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 group (Rn, +) = R x R x ··· x R. for n factors, which consists of ordered n-tuples under addition by
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 field F if algebraic over F if and only if α is a zero of some polynomial.
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 extension. ___ c. Every algebraic extension of a field is a finite extension. ___ d. The top field
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 extension of F, and [K : F] = [K : E][E : F]Proof Let {αi | i = 1, ··· , n} be a basis for E as
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 field F is transcendental over F if and only if β is not a zero of any polynomial in F[x].
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, α is algebraic over F. By Theorem 30 .19 if [E : F] = n, then 1, α,··· ,αn cannot be linearly
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 a linear combination of the βi.
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 that satisfy all m equations, if and only if the vector β = (b1 , ··· , bm) of Fm is a
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 coefficients equal to 1.
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 {βi | i ∈ I} is a basis for V over F, show that a linear transformation ∅ : V → V' is
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 F if and only if there exists some α ∈ E such that no proper subfield of E contains α.
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 space Fn of ordered n-tuples of elements of F over the field F, for any field F. What is a basis for
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 extension field of Q.____ e. Q is an extension field of Z2.____ f. Let α ∈ C be algebraic over Q of
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 algebraic of degree 5 over E.
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 does the concept of a linear transformation correspond? b. Define the kernel (or nullspace) of
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? Argue from this that a zero of x14 - 3x2 + 12 over Q can never be expressed as a rational function
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 (Z2(α))[x] by actually finding this factorization.
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 abelian groups. As usual, (Z2(α))* is the set of nonzero elements of Z2(α).
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 transformation. a. Show that ∅[VJ is a subspace of V'. b. Show that dim(∅[V]) = dim(V) - dim(Ker(∅)).
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 appropriate?
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 an α ∈ E such that f(α) = 0.Proof By Theorem 23.20, f(x) has a factorization in F[x] into
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 p2 elements for every prime p in Z+.
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 field F, where F has q elements. Let α ∈ E be algebraic over F of degree n. Prove that F(α) has qn
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 abelian group (Z x Z, +) discussed in Section 11. It is straightforward to verify that two elements of
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 from in Example 18.15 We claim that for integers r ands where gcd(r, s) = 1, the rings Zrs and Zr
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) = 1 for i ≠ j. Then there exists x ∈ Z+ such that x = ai (mod bi) for i = 1, 2, ···· ,
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 identity element, a(b + c) =(ab)+ (ac) and (a+ b)c = (ac) + (be) for all a, b, c ∈ S. Show that
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 connection between this exercise and Example 26.12.Data from Example 26.12Let F be the ring of all
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 + 3yz3
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] ⊆ N'. (Use Exercise 39 of Section 14.)Data from Exercise 39 of Section 14Let G and G' be groups, and
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 multiplicative group { 1, -1}. What is the kernel? Compare with Example 13.3.Data from example 13.313.3
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 ∅(ai) = µ(ai) for every i ∈ I. Prove that ∅ = µ. [Thus, for example, a homomorphism of a
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 Example An interesting example for us is the group S3 of3! = 6 elements. Let the set A be { 1, 2, 3}. We
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 a subfield of a field E. let a be any element of E, and let x be an indeterminate. The map ∅α :
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 < of Theorem 25.5. a b.c.d.e. 4x Data from Example 25.9 (Formal Laurent
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 indeterminate. The map ∅α : F [x] → E defined by ∅α(a0 + a1x + anxn) = a0 + a1α + .....
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 < of Theorem 25.5.a.b.c.d.e.Data from Example 25.9 (Formal Laurent Series Fields)
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 < of Theorem 25.5.a.b.c.d.e.Data from Example 25.9 (Formal Laurent Series Fields)
The polynomial 2x3 + 3x2 - 7x - 5 can be factored into linear factors in Z11[x ]. Find this factorization.

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