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mathematics
a first course in abstract algebra

Questions and Answers of A First Course In Abstract Algebra

We consider the field E = Q(√2, √3, √5). It can be shown that [E : Q] = 8. In the notation of Theorem 48.3, we have the following conjugation isomorphisms (which are here automorphisms of
Let ζ be a primitive 5th root of unity in C.a. Show that Q(ζ) is the splitting field of x5 - 1 over Q.b. Show that every automorphism of K= Q(ζ) maps ζ onto some power ζr of ζ. c. Using
In the easiest way possible, describe the group of the polynomial (x8 - 1) ∈Q[x] over Q.
Let f(x) ∈ F[x] be a monic polynomial of degree n having all its irreducible factors separable over F. Let K ≤ F̅ be the splitting field of f(x) over F, and suppose that f(x) factors in K[x]
Mark each of the following true of false.___ a. Let F be a field of characteristic 0. A polynomial in F[x] is solvable by radicals if and only if its splitting field in F̅ is contained in an
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.|λ(K)|Data from
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.|λ(Q(√2 +
By referring to Example 46.11, actually express the gcd 23 in the form λ(22,471) + µ,(3,266) for λ,µ, ∈ Z.Data from in Example 46.11Note that the division algorithm Condition 1 in the
let R[x] have the ordering given by i. Plow ii. PHigh as described in Example 25.2. In each case (i) and (ii), list the labels a, b, c, d, e of the given polynomials
F = E = Z7 in Theorem 22.4. Compute for the indicated evaluation homomorphism. ∅3[(x4 + 2x)(x3 - 3x2 + 3)]Data from Theorem 22.4Let F be a subfield of a field E. let a be any element of E, and
Show that x4 + 1 is irreducible in Q[x], as we asserted in Example 54.7.Data from in 54.7 Example Consider the splitting field of x4 + 1 over Q. By Theorem 23.11, we can show that x4 + 1 is
let R[x] have the ordering given by i. Plow ii. PHigh as described in Example 25.2. In each case (i) and (ii), list the labels a, b, c, d, e of the given polynomials
Find a gcd of x10 - 3x9 + 3x8 - 11x7 + 11x6 - 11x5 + 19x4 - 13x3 + 8x2 -9x + 3 and x6 - 3x5 + 3x4 - 9x3 + 5x2 - 5x + 2 in Q[x].
Let y and z be indeterminates, and let u = y¹² and v = z18. Describe the separable closure of Z3 (u, v) in Z3(y, z).
For the given isomorphic mapping of a subfield of E, give all extensions of the mapping to an isomorphic mapping of E onto a subfield of Q̅. Describe the extensions by giving values on the
Can the splitting field K of x2 + x + 1 over Z2 be obtained by adjoining a square root to Z2 of an element in Z2? Is K an extension of Z2 by radicals?
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.{K : Q}Data from
Compute the given arithmetic matrix expression, if it is defined. -2 -2 4 1 5 [9 1 + 4 -3 2
Referring to Example 55.3, complete the indicated computation, showing that Φ8(x) = x4 + 1.
Find the degree over Q of the splitting field over Q of the given polynomial in Q[x].x2 + 3
Find α such that the given field is Q(α). Show that your α is indeed in the given field. Verify by direct computation that the given generators for the extension of Q can indeed be expressed as
Find all conjugates in C of the given number over the given field.√2 over Q
Let and z be indeterminates, and let u = y¹² and v = y²z18. Describe the separable closure of Z3(u, v) inZ3(y, z).
Verify that the intermediate fields given in the field diagram in Fig. 54.6 are correctFigure 54.6 Field diagram - Q(V/2) = KH₂ Q(i√/2) = KHs Q(√2) = KH₁ Q(√2, 1) = K = Klat Q(√2, 1) =
For the given isomorphic mapping of a subfield of E, give all extensions of the mapping to an isomorphic mapping of E onto a subfield of Q̅. Describe the extensions by giving values on the
Is every polynomial in F[x] of the form ax8 + bx6 + cx4 + dx2 + e, where a ≠ 0, solvable by radicals over F, if F is of characteristic 0? Why or why not?
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.|G(K/Q)|Data from
Classify the group of the polynomial (x20 - 1) ∈ Q[x] over Q according to the Fundamental Theorem of finitely generated abelian groups.
Compute the given arithmetic matrix expression, if it is defined. +i -2 [¹# 73=30+3² [3 4 i 2-i i −2+ 17²+1 1] i 0 i
Find the degree over Q of the splitting field over Q of the given polynomial in Q[x].x4 - 1
For each field in the field diagram in Fig. 54.6, find a primitive element generating the field over Q.Figure 54.6 field diagram Q(i√2) = K₁0₁.03) Q(¹ + ¹) = K Q(i) = K (0₁.0₂) Q =
Find α such that the given field is Q(α). Show that your α is indeed in the given field. Verify by direct computation that the given generators for the extension of Q can indeed be expressed as
Find all conjugates in C of the given number over the given field.√2 over R
Referring to Exercise 1, describe the totally inseparable closure (see Exercise 6) of Z3(u, v) in Z3(y, z).Data from Exercise 1Let y and z be indeterminates, and let u = y¹² and v = z18. Describe
For the given isomorphic mapping of a subfield of E, give all extensions of the mapping to an isomorphic mapping of E onto a subfield of Q̅. Describe the extensions by giving values on the
Compute the given arithmetic matrix expression, if it is defined. i 4 3 1 -2i 3-i 2 3 4i 1 + i -i
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.|λ(Q)|Data from
Let F be a field, and let f(x) = ax2 + bx + c be in F[x], where a ≠ 0. Show that if the characteristic of F is not 2, the splitting field of f (x) over F is F(√b² - 4ac)
It is a fact, which we can verify by cubing, that the zeros of x3 - 2 in Q arewhere 3√2, as usual, is the real cube root of 2.Describe all extensions of the identity map of Q to an isomorphism
Using the formula for φ(n) in terms of the factorization of n, as given in Eq. (1), compute the indicated value:a. φ(60) b. φ(l000) c. φ(8100)
Find the degree over Q of the splitting field over Q of the given polynomial in Q[x].(x2 - 2)(x2 - 3)
Find α such that the given field is Q(α). Show that your α is indeed in the given field. Verify by direct computation that the given generators for the extension of Q can indeed be expressed as
Find all conjugates in C of the given number over the given field.3 +√2 over Q
Compute the given arithmetic matrix expression, if it is defined. 3 24 3 -1
Referring to Exercise 2, describe the totally inseparable closure of Z3(u, v) in Z3(y, z). (See Exercise 6.)Data from Exercise 2Let and z be indeterminates, and let u = y¹² and v = y²z18. Describe
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.|λ(Q(√2,
Give the first 30 values of n ≥ 3 for which the regular n-gon is constructible with a straightedge and a compass.
Find the degree over Q of the splitting field over Q of the given polynomial in Q[x].x3 - 3
Find α such that the given field is Q(α). Show that your α is indeed in the given field. Verify by direct computation that the given generators for the extension of Q can indeed be expressed as
It is a fact, which we can verify by cubing, that the zeros of x3 - 2 in Q arewhere 3√2, as usual, is the real cube root of 2.Describe all extensions of the identity map of Q to an isomorphism
Find all conjugates in C of the given number over the given field.√2 - √3 over Q
Find all conjugates in C of the given number over the given field.√2 + i over Q
Mark each of the following true or false.___ a. No proper algebraic extension of an infinite field of characteristic p ≠ 0 is ever a separable extension.___ b. If F(a) is totally inseparable over F
Show that if F is a field of characteristic different from 2 and f(x) = ax4 + bx2 + c, where a ≠ 0, then f(x) is solvable by radicals over F.
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem 53.6.|λ(Q(√6))|Data
Compute the given arithmetic matrix expression, if it is defined. 3 1 5 -3 -4 2 2 1 6 非
Describe the group of the polynomial (x5 − 2) ∈ (Q(ζ))[x] over Q(ζ), where ζ is a primitive 5th root of unity.
Show that for a finite group, every refinement of a subnormal series with abelian quotients also has abelian quotients, thus completing the proof of Lemma 56.3.Data from Lemma 56.3Let F be a field of
Find the smallest angle of integral degree, that is, 1°, 2°, 3°, and so on, constructible with a straightedge and a compass.
Find the degree over Q of the splitting field over Q of the given polynomial in Q[x].x3 - 1
It is a fact, which we can verify by cubing, that the zeros of x3 - 2 in Q arewhere 3√2, as usual, is the real cube root of 2.Describe all extensions of the automorphism ψ√3,-√3 of
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.Let F̅ be an algebraic closure of a field F.
Find all conjugates in C of the given number over the given field.√2 + i over R
Compute the given arithmetic matrix expression, if it is defined. [4 - 3 1 -1 3 07 7 1
Show that if E is an algebraic extension of a field F, then the union of F with the set of all elements of E totally inseparable over F forms a subfield of E, the totally inseparable closure of F in
Let K be the splitting field of x12 - 1 over Q.a. Find [K : Q].b. Show that for σ ∈ G (K/Q), σ2 is the identity automorphism. Classify G(K/Q) according to the Fundamental Theorem 11.12 of
Find all conjugates in C of the given number over the given field. √1+√2 over Q
The field K = Q(√2, √3, √5) is a finite normal extension of Q. It can be shown that [K : Q] = 8. Compute the indicated numerical quantity. The notation is that of Theorem
Show that for a finite group, a subnormal series with solvable quotient groups can be refined to a composition series with abelian quotients, thus completing the proof of Theorem 56.4.Data from
Repeat Exercise 4 for ζ a primitive 7th root of unity in C.Data from exercise 4Let ζ be a primitive 5th root of unity in C.a. Show that Q(ζ) is the splitting field of x5 - 1 over Q.b. Show that
Find the degree over Q of the splitting field over Q of the given polynomial in Q[x].(x2 - 2)(x3 - 2)
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.Let F̅ be an algebraic closure of a field F.
Compute the given arithmetic matrix expression, if it is defined. [ 那 -2 1 4 i 3i 1 -2i
Show that a field F of characteristic p ≠ 0 is perfect if and only if FP = F, that is, every element of F is a pth power of some element of F.
Give an example of an f(x) ∈ Q[x] that has no zeros in Q but whose zeros in C are all of multiplicity 2.Explain how this is consistent with Theorem 51.13, which shows that Q is perfect.Data from
Find all conjugates in C of the given number over the given field. √1 + √2 over Q(√2)
Let σ be the automorphism of Q(π) that maps π onto -π.a. Describe the fixed field of σ.b. Describe all extensions of σ to an isomorphism mapping the field Q(√π) onto a subfield of Q̅(π)̅
This exercise exhibits a polynomial of degree 5 in Q[x] that is not solvable by radicals over Q.a. Show that if a subgroup H of S5 contains a cycle of length 5 and a transposition τ, then H = S5.b.
Let E be a finite extension of a field F of characteristic p. In the notation of Exercise 7, show that EP = E if and only if FP = F.
Refer to Example 50.9What is the order of G(Q(3√2, i√3)/Q)?Data from in Example 50.9 - Let 2 be the real cube root of 2, as usual. Now x³ 2 does not split in Q(V2), for Q(³2) < R and only one
Mark each of the following true or false.___ a. Let F(α) be any simple extension of a field F. Then every isomorphism of F onto a subfield of F̅ has an extension to an isomorphism of F(α) onto a
Compute the given arithmetic matrix expression, if it is defined. 1 4 可
Find the splitting field K in C of the polynomial (x4 - 4x² - 1) ∈ Q[x]. Compute the group of the polynomial over Q and exhibit the correspondence between the subgroups of G(K/Q) and the
We consider the field E = Q(√2, √3, √5). It can be shown that [E : Q] = 8. In the notation of Theorem 48.3, we have the following conjugation isomorphisms (which are here automorphisms of
Express each of the following symmetric functions in y1, y2, y3 over Q as a rational function of the elementary symmetric functions s1, s2, s3.a. y12 + y22 + y32b.
How many elements are there in the splitting field of x6 - 1 over Z3?
Mark each of the following true or false._______ a. Every finite extension of every field F is separable over F._______ b. Every finite extension of every finite field F is separable over
Compute the given arithmetic matrix expression, if it is defined. 4 1 i 1
Refer to Example 50.9What is the order of G(Q(3√2, i√3)/Q(3√2))?Data from in Example 50.9 - Let 2 be the real cube root of 2, as usual. Now x³ 2 does not split in Q(V2), for Q(³2) < R and
Let K be an algebraically closed field. Show that every isomorphism σ of K onto a subfield of itself such that K is algebraic over σ[K] is an automorphism of K, that is, is an onto map.
Describe the group of the polynomial (x4 - 1) ∈ Q[x] over Q.
We consider the field E = Q(√2, √3, √5). It can be shown that [E : Q] = 8. In the notation of Theorem 48.3, we have the following conjugation isomorphisms (which are here automorphisms of
Mark each of the following true or false.___ a. Фn(x) is irreducible over every field of characteristic 0.___ b. Every zero in c of Фn(x) is a primitive nth root of unity.___ c. The group of Фn(x)
Show that if F is a field of characteristic not dividing n, thenin F[x], where the product is over all divisors d of n. x" -1=Π®«(x) din
Show that if α, β ∈ F̅ are both separable over F, then α ± β, αβ, and α/β, if β ≠ 0, are all separable over F.
Let E be an algebraic extension of a field F. Show that every isomorphism of E onto a subfield of F leaving F̅ fixed can be extended to an automorphism of F̅.
Give the order and describe a generator of the group G(GF(729)/GF(9)).
Let α1, α2, α3 be the zeors in C of the polynomial (x3 - 4x2 + 6x - 2) E Q[x]. Find the polynomial having as zeros precisely the following:a. α1 + α2 + α3b. α12, α22, α32
Let K be the splitting field of x3 - 2 over Q.a. Describe the six elements of G(K/Q) by giving their values on 3√2 and i√3. (By Example 50.9, K =Q(3√2, i √3).)b. To what group we have seen
Let α be a zero of x3 +x2 + 1 overZ2. Show that x3 +x2 + 1 splits in Z2(α).

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