There are.... Polynomials of degree atmost n in the polynomial ring Z, (x O none O5+5^n O 5^(n+1) O 5^n

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

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

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Q: Show that f:Z2 -> Z2 defined by f(n) = n2-n is a ring homomarphism.

A: The element in Z2 are {0,1} Now, f(0) = 02 - 0 = 0           f(1) = 12 - 1 = 0 Hence, f maps Z2 to…

Q: Use mathematical Inductlon to prove that for r>0 and any positive Integer

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Q: Show that the greater than or equal re lation >) iš a partially order on the Set of integer.

A: Partially Ordered Relation: A relation R on set P is said to be partially ordered if and only if R…

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

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Q: Show that the cubic field generated by a root of f(x) = 3x2 base. Use the fact that…

A: From the given information. The function f(x) is as follows.

Q: Let (?,+, ⋅) be a ring with additive identity 0. Prove that for all x∈?, 0⋅x=0 and x⋅ 0 = 0.

A: We know that if (R,+,.) is a ring then (R,+) is an abelian group. And in abelian group, cancellation…

Q: Prove in an integral domain, a prime element is always irreducible.

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Q: 8. Show that an element x of an inner product space X cannot have "too many" Fourier coefficients…

A: Given: Inner product space X. To find: nm<m2x2

Q: What are the possible orders of the elements of the finite field F25 of 25 elements ?

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Q: Consider the ring R = (0,2,4,6, 8, 10} under addition and multiplication modulo 12. The char (R) is

A: Characteristic of a Ring R: Char(R) is n. Where n is smallest number such that r+r+..+r(n…

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

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Q: 2. Let F be a field of order 16 and d,,d,,d, denotes the number of elements of multiplicative order…

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Q: The ring Z[x]/ is: Not Integral domain Integral domain but not Field O None of these O Field

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

Q: Suppose that you have 3 and 8 cent stamps, how much postage can you create using these stamps? Prove…

A: We shall show that every postage >=14 can be created using 3 and 8 cent stamps. And then deduce…

Q: Show that 7 is irreducible in the ring Z[V5].

A: Consider the ring ℤ5. Any element of the above ring is of the form a+b5 where a and b are integers.…

Q: Let S = {( ) laeR}. Then S is a Field

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Q: Show that if x, y,z are positive integers then: z-gcd (x,y) = gcd(zx ,zy)

A: The greatest common divisor (gcd) is the largest positive integer that divides each of the integers…

Q: 2. Define reducible and irseducible elements ring R and irreducible elements in Zu, Zq nand Zio R…

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Q: /6. Let (R, +,) be an integral domain and consider the set Z1 of all integral multiples of the…

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Q: Find all possible ring homomorphisms o : Z10 --->Z15 (b) In each case identify the Kern

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Q: Find the order of [5] in Z23 using Fermat's Theorem.

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Q: Show that the ring (2ot) is principal idenl ring.

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Q: How many ring homomorphisms Z/35→Z/7  are there?

A: Given that ℤ35→ℤ7

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

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

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: Prove that the number i5 is not reversible in the ring Z[V-5]

A: Here we show that isqrt(5) is not reversible in the ring Z[sqrt(-5)].

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

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Q: Prove that a field with 16 elements contains a subfield with 4 elements.

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Q: The ring Zpg²r has exactly-- ---maximal ideals 2 1 4

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

Q: Determine the possible finite fields whose largest proper subfieldis GF(25).

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Q: There are Polynomials of degree atmost n in the polynomial ring Z3[x]. .... none 3 + 3^n 3^n 3^(n+1)

A: Option D

Q: prove in a field with four elements, F = {0,1,a,b}, that 1 + 1 = 0.

A: The given four elements are F = {0, 1, a, b}.Consider ab belongs to F and then there are four…

Q: There are. Polynomials of degree atmost n in the polynomial ring Z, (x). *** O 5+5n O none O5n O…

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

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Q: 2. Suppose F is a finite field. Prove that F× is cyclic. Deduce that x² = -1 has a solution in a…

A: Here, F× denotes F-0. We need to prove F× is a cyclic group under multiplication.

Q: There are ... Polynomials of degree atmost n in the polynomial ring Z3[x]. none 3^n O 3^(n+1) O 3 +…

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Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: Let x, y, z be elements of the real numbers. Use the field axioms and ordered axioms to prove -0=0.

A: We make use of the following field axioms:

Q: The ring Z[x]/ is: O Integral domain but not Field O Not Integral domain Field

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

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

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Q: I need the answer as soon as possible

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Q: 8 Every quotient of a principal ideal domain is a principal ideal domain. 9. Every integral domain…

A: Given: 8.Every quotient of a principal ideal domain is a principal ideal domain

Q: (1) Show that there are infinitely many irreducible polynomials in F[].

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Q: 4) In a ring R; The sum of two non-trivial idempotent elements is not always an if we take + = is…

A: 4) We need to show that , sum of two non trivial idempotent elements is not always idempotent. We…

Q: Abstract Algebra

A: To prove the existence of infinitely many monic irreducible polynomials over any given field F.

Q: (B) Define the integral domain ring. Is the product of integral domain rings also an integr domain?

A: We will define integral Domain ring.

Q: Every ring without zero divisor it is an integral domain. T OF O *

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Q: 5. Show that there are infinitely many irreducible polynomials in F,[X].

A: By the method of contradiction i have solved the problem