Please answer viii, ix and x 1.Which of the following statements are true? Give reasons for your answers. Marks willonly be given for valid justification of your answers.i)In a non-abelian group of order 27, the identity conjugacy class is the only class witha single element.ii)A finite field with 16 elements has a subfield with 8 elements.The number of distinct abelian groups of order p"p"p, are distinct primes and n, e N.iii)* is n,n,...n,, where theiv)If G is a finite group, such that Z(G) = G, then o(G) is a prime.v)If X is a G-set, G a group, and YCX is G-invariant, then X\Y is G-invariant.vi) Any group of order 202 is simple.vii) SL,(R) C0(3).viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R.ix) Every prime element of Z[x,,x,...,X,] is irreducible.х)|Aut (L/K)| = |Aut (L)| –|Aut(K).

Given :- (ix) Every prime element of ℤx1 , x2 ,x3 ,x4,........,xn is irreducible. (viii) If ℝ is…

Answered: viii) If R is an integral domain, then…

viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R. x) Every prime element of Z[x,X2,...,x,] is irreducible.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 11E: Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a...

See similar textbooks

Related questions

Q: The number of irreducible monic polynomials of degree 2 over Z5

Q: Prove that subring R[x2, x3] not isomorphic to R[x].

Q: Show that a nonconstant polynomial from Z[x] that is irreducibleover Z is primitive.

Q: Prove that Zo is not isomorphic to Z, X. 10 12

Q: 23. Prove that if D is an integral domain, then D[x] is also an integral domain.

Q: 17. Find a nontrivial proper ideal of Z x Z that is not prime.

Q: Prove that the only idempotent elements in an integral domain R with unity are 0 and 1.

A: Proof. Let x ∈ R be idempotent,

Q: In the domain of Gaussian integers Z[i], the element 23 is * Irreducible Reducible Unit None of the…

A: (i) Every prime is irreducible in Z[i].

Q: Let w be a primitive 12th root of unity over Q. Find the minimal polynomial for wª over Q.

Q: 6) The Ying of integers(2) is ideal of veal number (R) an DTrue D Fulse

Q: The ring Z,2, has exactly-- --maximal ideals O 2 3.

Q: Let i, j be coprime positive integers, and let R be an integral domain. Prove that (r-y) is a prime…

A: To show that the ideal generated by x^i-y^j under the given conditions is a prime ideal in the ring…

Q: If D is an integral domain and if na = 0 for some a is non zero in D and some non zero integer n,…

Q: ind subrings R1 and R2 of Z such that R1 union R2 is not a subring of Z

Q: Show that N= (3x, y)|x, y ∈ Z is a maximal ideal ofZ×Z. Note that you must show that it is indeed…

Q: (c) Show that the ideal generated by x² + y² + z² € C[x, y, z] is a prime ideal.

A: If <x2+y2+z2>is not a prime ideal, there should exist g,h∈ℝx, y, z s.t. x2+y2+z2 | gh,…

Q: Prove or disprove that if D is a principal ideal domain, then D[x] isa principal ideal domain.

A: Let the principal domain be D.

Q: 6. Consider the ring of polynomials with rational numbers as coefficients, Q[x]. Set R = {f(x) E…

Q: 9.16. Let R be a ring and I a proper ideal. 1. If R is an integral domain, does it follow that R/I…

A: Let R be a ring and I be an ideal. 1. Choose R=(ℤ, +, ·) and I=4ℤ. Result: Ideals of ℤ are nℤ where…

Q: 16. If R is a field, show that the only two ideals of R are {0} and R itself.

Q: (2) Every ring without zero divisor it is an integral domain. T O F

A: A domain is a ring with identity which is without any zero divisors. An integral domain is…

Q: (W/) Express In x in terms of In x, In y, and In z. - In y + 1. In z 2 2 In x + 1 2 In x- In y + In…

Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: 1. Find an r> 0 such that all zeroes of 24 + 6z²+z+3 are contained in D,(0).

Q: 20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 –…

A: Proved by using Rouches theorem from complex analysis.

Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

Q: Prove that Z[2√-3] is not a principal ideal domain

Q: Prove that if D is an integral domain, then D[x] is also an integral domain.

Q: (7) Is every rational function on ID holomorphic on D?

Q: 7. The integral domain (Z, +,.) can be embedded in the field (a) (Q, +,.) (b) (R, +,.) (c) (C,+,.)…

A: We know that every integral domain can be embedded in a field

Q: Let be R an integral domain and let / be an ideal in R, then R/Iis also an integral domain. Select…

Q: R/H is integral domain, if and only if H is.........of R. O (i) maximal ideal O (ii) prime ideal O…

A: Given: Suppose that R is an integral domain. Then R has at least two elements (since 1≠0) and hence…

Q: Prove that In( +VI-X²³) = In ( ¹+ √1-x²) sech ¹(x) =

Q: 6) let fl2) = CosCitaiy) 22atey Express flz) that f satisfies in the form and fornd z So the Canchy-…

Q: *).). (a) Show that X is measurable with respect to o-field G if and only if o(X) C G. Show that X…

Q: Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of…

Q: Use Eisenstein Criterion to verify that , If p = 2, f(x) = x² – 2 is irreducible %3D over Q. State…

A: Explanation of the solution is given below...

Q: 4Z is prime ideal of 2Z, but it is not Maximal ideal of Z. To fo

Q: - Let R=(Z9,+9,9). Find 1. Char(R) 2. Nilpotent ideals of R 3. Prime ideals of R.

A: Solution

Q: The union of two ideals is ideal* The only idempotent in an integral domain is 0 O T O F O O

Q: 1. An integral domain D is called a principal ideal domain if every ideal of D has the form (a) =…

A: Using Definition of Principal Ideal Domain we have to prove that Z is a Principal Ideal Domain.

Q: The ring Z,2, has exactly--- --maximal ideals 1 2.

Q: In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer). 1) Prove that I is an ideal of Z[x]. 2)…

Q: 2.3 Theorem. Let p(x) be an irreducible polynomial in F[x). Then there exists an extension E of F in…

Q: Let R = Z3[x]and let I = {q(x)(x² + 1)|q(x) E Z3[x]} then I is O Maximal ideal of R O Not prime…

Q: Prove that the set of all polynomials whose coefficients are all evenis a prime ideal in Z[x].

A: Assume A to be the subset of X[x] with all even coefficients.For∑i=0maixi∑j=0nbjxj=∑i=0maxm,nai−bixi…

Q: 12. Show that the set of all constant polynomials in Z[r] is a subring but not an ideal of Z[x]

Q: Let x and y be coprime positive integers. Prove gcd(x + y, x² + y²) E {1, 2}.

Q: Prove that I= (2+ 2i) is not a prime ideal of Z[1].

A: It is given that, I=2+2i is not a prime ideal of ℤi. Definition used: A prime ideal is an ideal I…

Question

Please answer viii, ix and x

1.
Which of the following statements are true? Give reasons for your answers. Marks will
only be given for valid justification of your answers.
i)
In a non-abelian group of order 27, the identity conjugacy class is the only class with
a single element.
ii)
A finite field with 16 elements has a subfield with 8 elements.
The number of distinct abelian groups of order p"p"
p, are distinct primes and n, e N.
iii)
* is n,n,...n,, where the
iv)
If G is a finite group, such that Z(G) = G, then o(G) is a prime.
v)
If X is a G-set, G a group, and YCX is G-invariant, then X\Y is G-invariant.
vi) Any group of order 202 is simple.
vii) SL,(R) C0(3).
viii) If R is an integral domain, then R/I is an integral domain, for any ideal I of R.
ix) Every prime element of Z[x,,x,...,X,] is irreducible.
х)
|Aut (L/K)| = |Aut (L)| –|Aut(K).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Pyramids$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
25. Prove or disprove that every group of order is abelian.
If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.
10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .
15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.
Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.
6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.