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mathematics
a first course in abstract algebra

Questions and Answers of A First Course In Abstract Algebra

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
F = E = Z7 in Theorem 22.4. Compute for the indicated evaluation homomorphism.∅4(3x106 + 5x99 + 2x53)Data from Theorem 22.4Let F be a subfield of a field E. let a be any element of E, and let
Prove from Exercise 12 that every nonzero commutative ring containing an element a that is not a divisor of 0 can be enlarged to a commutative ring with unity. Data from Exercise 12Let R be a
With reference to Exercise 12, describe the ring Q(Z, {2n |n ∈ Z+}), by describing a subring of R to which it is isomorphic.Data from Exercise 12Let R be a nonzero commutative ring, and let T be a
Demonstrate that x3 + 3x2 - 8 is irreducible over Q.
Let ∅a : Z5[x] → Z5 be an evaluation homomorphism as in Theorem 22.4. Use Fermat's theorem to evaluate ∅3(x231 + 3x117 - 2x53 + 1).Data from Theorem 22.4(The Evaluation Homomorphisms
Referring to Example 24.3, show that YX - XY = 1. Data from in 24.3 Example Let F be a field of characteristic zero, and let (F[x], +) be the additive group of the ring F[x] of polynomials
Demonstrate that x4 - 22x2 + 1 is irreducible over Q.
Referring to Example 25.12, show that the map ∅: Z[√2] → R where ∅(m + n√2) = m - n√2 is a homomorphism.Data from Example 25.12Exercise 11 of Section 18 shows that {m + n√2| m, n ∈ Z}
Let F be a field. Give five different characterizations of the elements A of Mn(F) that are divisors of 0.
Give a one-sentence synopsis of the proof of Theorem 20.8.Data from 20.8 Theorem (Euler's Theorem) If a is an integer relatively prime to n, then aφ(n) - 1 is divisible by n, that is, aφ(n) = 1
An element a of a ring R is nilpotent if an = 0 for some n ∈ Z+. Show that if a and b are nilpotent elements of a commutative ring, then a + b is also nilpotent.
Show that a ring R has no nonzero nilpotent element if and only if 0 is the only solution of x2 = 0 in R.
Show that a subset S of a ring R gives a subring of R if and only if the following hold: 0 ∈S; (a - b) ∈ S for all a, b ∈ S; ab ∈ S for all a,b ∈S.
a. Show that an intersection of subrings of a ring R is again a subring of R. b. Show that an intersection of subfields of a field F is again a subfield of F.
Let R be a ring, and let a be a fixed element of R. Let Ra be the subring of R that is the intersection of all subrings of R containing a. The ring Ra is the subring of R generated by a. Show that
Let R be a ring, and let a be a fixed element of R. Let Ia = {x ∈ R | ax = 0}. Show that Ia is a subring of R.
Find all positive integers n such that Zn contains a subring isomorphic to Z2.
Write the polynomials in R[x, y, z] in decreasing term order, using the order lex for power products xmynZs where z < y < X.3y2z5 - 4x + 5y3z3 - 8z7
Find all ideals N of Z12. In each case compute Z12/N ; that is, find a known ring to which the quotient ring is isomorphic.
Write the polynomials in R[x, y, z] in decreasing term order, using the order lex for power products xmynZs where z < y < X.3y - 7x + 10z3 - 2xy2z2 + 2x2yz2
Find all prime ideals and all maximal ideals of Z2 x Z2.
Give addition and multiplication tables for 2Z/8Z Are 2Z/8Z and Z4 isomorphic rings?
Write the polynomials in R[x, y, z] in decreasing term order, using the order lex for power products xmynZs where z < y < X.38 - 4xz + 2yz - 8xy + 3yz3
Find all prime ideals and all maximal ideals of Z2 x Z4.
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 isomorphism of a ring R with a ring
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 1. Data from Exercise 12xy3z5 -
Find all c ∈ Z3 such that Z3[x]/(x2 + c) is a field.
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 ideal N of a ring R is an additive
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 2.Data from Exercise 23y2z5 - 4x + 5y3z3 -
Find all c ∈ Z3 such that Z3[x]/(x3 + x2 + c) is a field.
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 kernel of a homomorphism ∅
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 3.Data from Exercise 33y - 7x + 10z3 -
Find all c ∈ Z3 such that Z3[x]/(x3 + cx2 + 1) is a field.
Mark each of the following true or false. ___ a. The concept of a ring homomorphism is closely connected with the idea of a factor ring. ___ b. A ring homomorphism ∅ : R → R' carries
Write the polynomials in order of decreasing terms using the order deglex with power products xmynzs where z < y < x.The polynomial in Exercise 1.Data from Exercise 12xy3z5 - 5x2yz3 + 7x2y2z -
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 maximal ideal of a ring R is an ideal
Let R be a ring. Observe that {0} and R are both ideals of R. Are the factor rings R/R and R/{0} of real interest? Why?
Write the polynomials in order of decreasing terms using the order deglex with power products xmynzs where z < y < x.The polynomial in Exercise 2.Data from Exercise 23y2z5 - 4x + 5y3z3 -
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 prime ideal of a commutative ring R is an
Give an example to show that a factor ring of an integral domain may be a field.
Write the polynomials in order of decreasing terms using the order deglex with power products xmynzs where z < y < x.The polynomial in Exercise 3.Data from Exercise 33y - 7x + 10z3 - 2xy2z2 +
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 prime field is a field that has no proper
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 principal ideal of a commutative ring with
Mark each of the following true or false. ___ a. Every prime ideal of every commutative ring with unity is a maximal ideal. ___ b. Every maximal ideal of every commutative ring with unity
Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain.
Let power products in R[x, y, z] have order lex where z < y < x. If possible, perform a single-step division algorithm reduction that changes the given ideal basis to one having smaller maximum
Find a maximal ideal of Z x Z.
Find a subring of the ring Z x Z that is not an ideal of Z x Z.
Let R = {a + b√2 | a, b ∈ Z} and let R' consist of all 2 x 2 matrices of the formShow that R is a subring of !R. and that R' is a subring of M2(Z). Then show that ∅ : R → R', where ∅(a +
Let power products in R[x, y, z] have order lex where z < y < x. If possible, perform a single-step division algorithm reduction that changes the given ideal basis to one having smaller maximum
Find a prime ideal of Z x Z that is not maximal.
A student is asked to prove that a quotient ring of a ring R modulo an ideal N is commutative if and only if (rs - sr) ∈ N for all r, s ∈ R. The student starts out: Assume R/ N is commutative.
Let power products in R[x, y, z] have order lex where z < y < x. If possible, perform a single-step division algorithm reduction that changes the given ideal basis to one having smaller maximum
Find a nontrivial proper ideal of Z x Z that is not prime.
Let power products in R[x, y, z] have order lex where z < y < x. If possible, perform a single-step division algorithm reduction that changes the given ideal basis to one having smaller maximum
Is Q[x]/(x2 - 5x + 6) a field? Why?
Show that each homomorphism from a field to a ring is either one to one or maps everything onto 0.
Let the order of power products in R[w , x, y, z] be lex with z < y < x < w. Find a Grobner basis for the given ideal.(w + x - y + 4z - 3, 2w + x + y - 2z + 4, w + 3x - 3y + z - 5)
Is Q[x]/(x2 - 6x + 6) a field? Why?
Show that if R, R', and R" are rings, and if ∅ : R → R' and ψ : R' → R" are homomorphisms, then the composite function ψ∅ : R → R" is a homomorphism. Data from Exercise 49 Show
Let the order of power products in R[w , x, y, z] be lex with z < y < x < w. Find a Grobner basis for the given ideal.(w - 4x + 3y - z + 2, 2w - 2x + y - 2z + 5, w - 10x + 8y - z - 1)
Give a one- or two-sentence synopsis of "only if' part of Theorem 27.9. Data from Theorem 27.9Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a
Let R be a commutative ring with unity of prime characteristic p. Show that the map ∅p : R → R given by ∅p(a) = aP is a homomorphism (the Frobenius homomorphism).
Find a Grobner basis for the indicated ideal in R[x].(x4 + x3 - 3x2 - 4x - 4, x3 + x2 - 4x - 4)
Give a one- or two-sentence synopsis of "if' part of Theorem 27.9.Data from Theorem 27.9Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a
Let Rand R' be rings and let ∅ : R → R' be a ring homomorphism such that ∅[R] ≠ {0'}. Show that if R has unity 1 and R' has no 0 divisors, then ∅(1) is unity for R'.
Find a Grobner basis for the indicated ideal in R[x].(x4 - 4x3 + 5x2 - 2x, x3 - x2 - 4x + 4, x3 - 3x + 2)
Give a one- or two-sentence synopsis of Theorem 27.24.Data from 27.24 Theorem If F is a field, every ideal in F[x] is principal. Proof Let N be an ideal of F[x]. If N = {0}, then N = (0).
Let ∅ : R → R' be a ring homomorphism and let N be an ideal of R. a. Show that ∅[N] is an ideal of ∅[R]. b. Give an example to show that ∅[N] need not be an ideal of R'. c.
Give a one- or two-sentence synopsis of the "only if" part of Theorem 27.25.Data from 27.25 Theorem An ideal (p(x)} ≠ {0} of F[x] is maximal if and only if p(x) is irreducible over
Find a Grobner basis for the indicated ideal in R[x].(x5 + x2 + 2x - 5, x3 - x2 + x - 1)
Let F be a field, and let S be any subset of F x F x • • • x F for n factors. Show that the set Ns of all f(x1, • • •, Xn) ∈ F[x1, • • •, xn]that have every element (a1,
Find a Grobner basis for the given ideal in R[x, y]. Consider the order of power products to be lex with y < x. If you can, describe the corresponding algebraic variety in R[x, y].(x2y - x - 2,xy
Let R be a finite commutative ring with unity. Show that every prime ideal in R is a maximal ideal.
Show that a factor ring of a field is either the trivial (zero) ring of one element or is isomorphic to the field.
Find a Grobner basis for the given ideal in R[x, y]. Consider the order of power products to be lex with y < x. If you can, describe the corresponding algebraic variety in R[x, y].(x2y + x, xy2 -
Corollary 27.18 tells us that every ring with unity contains a subring isomorphic to either Z or some Zn. Is it possible that a ring with unity may simultaneously contain two subrings isomorphic to
Show that if R is a ring with unity and N is an ideal of R such that N ≠ R, then R/N is a ring with unity.
Find a Grobner basis for the given ideal in R[x, y]. Consider the order of power products to be lex with y < x. If you can, describe the corresponding algebraic variety in R[x, y].(x2y + x + 1,
Continuing Exercise 25, is it possible that a ring with unity may simultaneously contain two subrings isomorphic to the fields Zp and Zq for two different primes p and q? Give an example or prove it
Let R be a commutative ring and let a ∈ R. Show that Ia = {x ∈ R | ax = 0} is an ideal of R.
Find a Grobner basis for the given ideal in R[x, y]. Consider the order of power products to be lex with y < x. If you can, describe the corresponding algebraic variety in R[x, y].(x2y + xy2, xy -
Following the idea of Exercise 26, is it possible for an integral domain to contain two subrings isomorphic to Zp and Zq for p ≠ q and p and q both prime? Give reasons or an illustration.Data from
Show that an intersection of ideals of a ring R is again an ideal of R.
Let F be a field. Mark each of the following true or false. ___ a. Every ideal in F[x] has a finite basis. ___ b. Every subset of R2 is an algebraic variety. ___ c. The empty subset of
Prove directly from the definitions of maximal and prime ideals that every maximal ideal of a commutative ring R with unity is a prime ideal.
Show that N is a maximal ideal in a ring R if and only if R/N is a simple ring, that is, it is nontrivial and has no proper nontrivial ideals. (Compare with Theorem 15.18.)Data from Theorem 15.18M is
Let ∅ be a homomorphism of a ring R with unity onto a nonzero ring R'. Let u be a unit in R. Show that ∅(u) is a unit in R'.
Show that if f1, f2, ···,fr are elements of a commutative ring R with unity, then I= {c1f1 + c2f2 + • • • + crfr|ci ∈ I for i = 1,···,r}is an ideal of R.
An element a of a ring R is nilpotent if an= 0 for some n ∈ Z+. Show that the collection of all nilpotent elements in a commutative ring R is an ideal, the nilradical of R.
Show that if f(x) = g(x)q(x) + r(x) in F[x], then the common divisors in F[x] of f(x) and g(x) are the same as the common divisors in F[x] of g(x) and r(x).
Let F be a field and f(x), g(x) ∈ F[x]. Show that f(x) divides g(x) if and only if g(x) ∈ (f(x)).
Referring to the definition given in Exercise 30, find the nilradical of the ring Z12 and observe that it is one of the ideals of Z12 found in Exercise 3. What is the nilradical of Z? of Z32?Data
Show that {xy, y2 - y) is a Grobner basis for (xy, y2 - y), as asserted after Example 28.9.Data from Example 28.9.By division, reduce the basis {xy2, y2 - y} for the ideal I = (xy2, y2 - y) in
Let F be a field and let f (x), g(x) ∈ F[x]. Show that N = {r(x)f(x) + s(x)g(x) | r(x), s(x) ∈ F[x]} is an ideal of F[x]. Show that if f(x) and g(x) have different degrees and N ≠ F[x], then
Referring to Exercise 30, show that if N is the nilradical of a commutative ring R, then R/N has as nilradical the trivial ideal {0 + N}.Data from Exercise 30An element a of a ring R is nilpotent if
Let F be a field. Show that if S is a nonempty subset of Fn, then I(S) = {f(x) ∈ F[x]|f(s) = 0 for all s ∈ S} is an ideal of F[x].
There is a sort of arithmetic of ideals in a ring. The exercises define sum, product, and quotient of ideals.Let A and B be ideals of a ring R. The product AB of A and B is defined by a. Show
Use Theorem 27.24 to prove the equivalence of these two theorems: Fundamental Theorem of Algebra: Every nonconstant polynomial in C[x] has a zero in C.Nullstellensatz for C[x]: Let f1(x),

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