2. If A + 4, N, what is the Borel field generated by {A}.

Q: Find the smallest field that has exactly 6 subfields.

A:

Q: 32. Suppose B1, B2 are o-fields of subsets of 2 such that B1 c B2 and B2 is countably generated.…

A:

Q: Prove that Q(√2) is a field

A:

Q: 10. Which of the following is not a field of quotients: (a) (Z2, +2,-2) (b) (Q, +,.) (c) (C,+,.) (d)…

A: Option (c) is correct

Q: Let S = {( ) laeR). Then S is a Field True False O O

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

A:

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: Let α =21/8 and let K = Q(α,i). Show that (a) K is the splitting field of x8 −2 over Q and dimQ(K) =…

A: Given: α=218 and K=Qα,i.   To show: K is a splitting field of x8-2 over ℚ and dimℚK=16

Q: 32. Suppose B1, B2 are o-fields of subsets of 2 such that B C B2 and B2 is countably generated. Show…

A:

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

A:

Q: 14. Let x and y be elements in a field F. If xy = 0, then either x = 0 or y = 0. * True False

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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

A:

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

A:

Q: It is possible to write z² + 6z + 4 in Zu[z] with the help of linear factors. Find this…

A: (a). The given polynomial is px=x2+6x+4 in ℤ11x. Observe that  p1=12+61+4=11≈0 Hence the polynomial…

Q: One of the following is not a field Z33 Q Z3[i]

A: We have to choose which one of the following is not a field among the given sets.

Q: 26 a ba Q(1E) = }a+bV2 / a, be Q4 2.1) Show (K,t,') eld under add. and rmult, of amotmias 2.2) Show…

A:

Q: 12) Consider he algebraic extension E = bQ CVE, B, VF) of The field Q of rahionat numbers. Then (E :…

A: Given E=Q(2,3,5) of the field Q of rational numbers.

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

A:

Q: Let f(x) € Q[x] \ Q and KC C be the splitting field of

A:

Q: Let S = {( ) laeR). Then S is a Field O True False

A: I have proved the all conditions for field.

Q: If 0±x#1 in a field R, then x is an idempotent. но чо

A: Only idempotent element in a field are 0 and 1

Q: If f is a field containing an in finte humber of destinct of distinst elen

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: Suppose that ℝ×ℝ, +, · is a field where (x, y) + (z, w) = (x+z, y+w) and x, y·z, w = xz-yw, xw+yz…

Q: Rp { -b a | a, b = Zp}

A: It is given that,  Rp = ab-ba : a, b ∈ ℤp We have to prove that,  Rp is a commutative ring with unit…

Q: Let S = JaeR}. Then S is a Field True False

A:

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: {0, A, Aº, $2} and X2 = {¢, B, Bº, S2} are o-fields, but &¡ U X2 is not. Why?

A:

Q: 15. If S1 and S2 are two semialgebras of subsets of 2, show that the class S1S2 := {A1A2 : A1 € S1,…

A:

Q: 5. Prove that Z[i]/(1 – i) is a field.

A:

Q: Suppose 2 = {0, 1, 2} and C = {{0}}. Enumerate N, the class of all o-fields containing C and give o…

A: We have to find the class of all sigma fields containing C.

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: a field and let c,d ∈ 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: Let S = {( ) laeR}. Then S is a Field O True O False

A:

Q: 6. Show that Q(V3) = {a + bv3 : a, b e Q} is a field.

A:

Q: 5: Let R=(Z,+, .). Find Chan Idempotent element of R e) Is R a field? Why? c) Nilpotent elements of…

A: Given: Let R=Z,+,· To find: Nilpotent elements and idempotent element of R. Is R a field? Why?…

Q: Consider the field Z3[x]/. Please explain to me how this field has 9 elements. Thank you!

A: Consider the field Z3[x]/<x2 +x+2>. Please explain to me how this field has 9 elements.

Q: 18. Is Q[x]/(x² – 5x + 6) a field? Why? 10

A:

Q: IN AN AC GENERATOR, WHICH CONTAINS A PAIR OF MAGNETS AND A COIL, IF THE ROTATING COIL IS IN…

A: True

Q: 2. Prove that F = {a+b√√3 | a,b ≤ R} is a field. Be sure to give a clear justification for each…

A: The given set is F=a+b3| a, b∈ℝ. Prove F is a field by showing it satisfies all the axioms.…

Q: 1. Suppose 2 containing C. {0, 1} and C {{0}}. Enumerate N, the class of all a-fields %3D %3D

A:

Q: If F = Za[z] (z²–cz+1) find the value of c for which F will be a field? %3D

A: F is field iff x2-cx+1 is irreducible over f.

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x +z, y + w) and (x, y). (z, w) =…

A:

Q: Q2:(2/Ptiti.) is a Prime field; Pprime ? a.

A: The given problem is related with prime field. Given that,                 ℤpℤ , + , . , where p is…

Q: Suppose that An are fields satisfying An C An+1. Show that Un An is a field. (But see also the next…

A:

Q: 1. Suppose that (Q x Q, +,.) is a field where (x, y) + (z, w) = (x + z,y + w) and (x, y). (z, w) =…

A:

Q: 1. Suppose that (R × R, +, .) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: I have proved (1, 0) is the identity element and the checked by option one by one

Q: [3 2 [1 4] 2) Consider the field F5. Let W = span C Max2(F5). Is [1 2] 1 e W? Your answer to this…

A:

Q: 1. Suppose that (R x R, +,.) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: To find the inverse element of (1, 2) first we find the identity element in the given field, and…

Q: 1. Prove that, if F is Borel field in 2, then (i) ø E F (ii) whenever A,,A2, ... E F, then also N1 A…

A: Definition of Borel field: Let Ω be a space. Let ℱ be a collection of subsets of Ω. Then ℱ is said…