Q. 8. Show that (Z,,+6,×6) is a commutative ring. Is (Z,,+s,×s) a field? Justify your answer.

Q: 18. Prove that in a Euclidean ring R, (a, b) can be found as follows : b= 90 a+ r,, where d (r) <d…

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Q: Prove that Q(√2) is a field

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Q: Need correct answer, Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

A: Solution:-

Q: For which value of k is the quotient ring Zs[x]/(x³+ 2x² + kx + 3) a field? 1 O 2 All of the above

A: Here it is given that: ℤ5xx3+2x2+kx+3 Here, p(x) is a polynomial of degree 3. If it has a zero ℤ5…

Q: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

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Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

Q: Show that Q(V5) is a field

A: We show that Q(√5) is a field.  Defintion:  A non trivial ring R with unity is a field if it be…

Q: Show by any means that Q[r]/(x4 + 3x2 + 9x + 1) is a field.

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Q: Zg /2Z, is a field Select one: O True O False

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Q: 4. Is M2(R) a field? Justify. -{[i !] 5. Show that T = | |x, y, z €R is a subring of M2(R).

A: Introduction: A ring is another important algebraic structure under two binary operations. These…

Q: In the ring Z;[x]': 1- The inverse of x is:

A: Answer :

Q: 4. Determine whether or not each of the following factor ring is a field. (a) Q[x]/{x² – 5x +6) (b)…

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Q: Show that Q(√7, i) is the splitting field for x4 - 6x2 - 7.

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Q: 19. Is Q[x]/(x² – 6x + 6) a field? Why?

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Q: Find the degree of the splitting field of x^2-3 over Q.

A: Degree of splitting field

Q: If ne Z not prime, then the ring (Zn, +ni'n) is a) Integral domain b) field c) Division ring d) not…

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Q: Consider the ring R = {r,s,t} whose addition and multiplications tables are given below. Then t.s =

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Q: Let (R, +, ) be a ring, a, b e R with a + 0. Then the equation ax = b has a solution in R.

A:

Q: In which of the following rings is the polynomial f(x) = 2x2 – 1 irreducible? Z[r] Q[e] Option 1…

A: Irreducible polynomial

Q: Is Q [x]/⟨x2 -5x + 6⟩ a field? Why?

A:

Q: 3. How many units are in the ring (Z/15Z) × (Z/125Z)?

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Q: 9. Suppose that (R,+, .) be a commutative ring with identity and xE rad R, then (a) (x) = R (b) 1-x…

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Q: The characteristic of the ring Z3XZ6 is 9. Select one: O 3 O None O 18 O 6

A: The characteristics of the ring is defined as  the least positive integer n such that na=0 for all…

Q: Show that Q(√7, i) is the splitting field for x4 - 6x2 -7.

A: x4 - 6x2 -7 = x4 - 7x2 + x2 -7                   = x2( x2 - 7) + 1 ( x2 - 7 )                   =…

Q: Q. 8. Show that (Z,,+6,×6) is a commutative ring. Is (Z,,+6,×6) a field? Justify your answer.

A: Note: As per bartleby instruction when more than one question is given only one has to be answered.…

Q: Show that 2Z ∪ 3Z is not a subring of Z.

A: Theorem for Sebring Test: For any ring R, a subset S of R is a subring if and only if: it is closed…

Q: 4. Is M₂ (R) a field? Justify. I 5. Show that T = = { [ ² ] 12,9₁² € R} i 2 is a subring of M₂ (R).

A: Subring of a ring

Q: Find the splitting field of x4 + 1 over Q.

A:

Q: Q Show that Z12/17) is a quatient ring.

A: We have to show that ,   ℤ12(4¯)  is a quotient ring.

Q: Which one of the following is true? In the ring Z6, if x = x then just x = 0 or x = 1 In the ring…

A: Option (3) is correct. In the ring Zp, if xy=0, then x=0 or y=0.

Q: 35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and…

A: The objective is to show that the first ring of the following is not isomorphic to the second:

Q: Which one of the following is true? In the ring Zp, if x = x then x = 0 or x = 1 In the ring Z6, 4…

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Q: For which value of k is the quotient ring Z5[x]/ (x^3 + 2x^2 +kx +3) a field?

A:

Q: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, US2 is subring of R iff either S,…

A: By supposinɡ S1 and S2 as two subrinɡs of rinɡ (R, +, .) To prove that S1∪ S2 is subrinɡ of R if and…

Q: handwritten solution asap for part b

A:

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…

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Q: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either S,…

A: I jave used the definition of subring

Q: and t=t+1}. nts in K? List all the elements in K. a field? e)rK is a field, draw two tables to prove…

A: This is a problem of Field Theory.

Q: Consider the ring (Q,Ð,0), where the "addition" O and "multiplication" ® are defined as follows:…

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Q: 18. Prove that in a Euclidean ring R, (a, b) can be found as follows : b= 90 a+ r,, where d (r) <d…

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Q: C. Prove that F2 (as defined on p20 of the notes) is a field.

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Q: Which one of the following is not a commutative ring? (a) (Q. +, :) (b) (Z5, O5, 85) (c) (Z12, O12,…

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Q: In the ring Z[x]/: 1- The inverse of x is:

A: The given ring is ℤ2x/x3+1. The objective is to find the inverse of x and to find Uℤ⊕ℤ.

Q: The number ofnilpotent elements in the ring Z23 is: 2 4 8 O 1

A: Here we need to find the number of nilpotent elements in Z_23  Since Z_23 is a field and it doesn't…

Q: 9. The number of ideals in the ring (Zis.+1s1s) is: (a) Two (b) Three (e) Four (d) Five.

A: The positive divisor of 15 are: 1,3,5,15

Q: The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +, .) (b) ((8),+,.) (c) ((12), +,.)…

A: Prime ideal sometimes behaves like a prime numbers . Let's firstly define prime ideal.

Q: (b) if R is commutative and has no ideals other than {0} and R, then R is a field.

A: 4.b) Given that R is commutative with with unity has no ideal other than {0} and R.

Q: Is the factor ring Z5[x]/ a field?

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Q: s{a+bV2:a, b e Z } under addition and multiplication a ring? Justify. Is it a mmutative ring?

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Q: Show that R[x]/&lt;x2 +1&gt; is a field.

A: To show that ℝx/x2 + 1 is a field, we enough to show that x2+1 is maximal in ℝx. Suppose that I =…