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

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

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Q: Use Eisenstein criterion to verify that, if p =2 , f(x)= x2 -2 is irreducible over Q, state the…

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

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Q: 4. Show that 7 is irreducible in the ring Z[V5] using the norm N defined by N(a + bv5) = | a? –…

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Q: Prove that [x] ∈ ℤ/n is a zero divisor if and only if there exists an integer y >1 such that y | x…

A: Let x∈ℤn is a zero divisor. There there exists a nonzero element y such that  xy=0 in ℤn. Hence xy…

Q: (a) Prove that every element of Q/Z has finite order.

A: Note:- As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: Prove that the ideal <x2 + 2> is prime in Z[x] but not maximal in Z[x].

A: The prime ideal <x2+2> is not maximal in Z[x] if Z[x]/<x2+2> is isomorphic to the…

Q: Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a…

A: Let [a]p be a nonzero element in Z/pZ.

Q: Prove that that the natural numbers with the binary operation of multiplication (as defined in the…

A: To prove that the natural numbers with the binary operation of multiplication forms a commutative…

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: Prove that Q is not isomorphic to Z.

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Q: Let A be an n × n symmetric negative definitematrix. Show that the leading principal submatrices of…

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Q: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

Q: Abstract Algebra. Please explain everything in detail.

A: To describe all the field automorphisms of the given field.

Q: Prove that R under addition is not isomorphic to R* under multiplication.

A: we know that R and R* = R – {0} are groups under addition and multiplication respectively. We have…

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: Show that there exist only two homomorphism from R into R?

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Q: 2. Prove that if 5BA+ AC A - I = A, then A must be inver

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Q: Prove that coth-la = In x +1 for |a| > 1 x - 1

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Q: Let S = {(°°) | VOER} s Show that S is subring of Max

A: Subring test : A non empty subset  S  of Ring  R is subring  of R iff S closed under subtraction and…

Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: 14. Let (a, b) = 1 and r = y® for some integers r and y. Then prove that there exists an %3D integer…

A: As per policy, we are solving only the first Question please repost 2nd Question to get its…

Q: Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors.

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Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

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Q: Show that 1 - i is an irreducible in Z[i].

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Q: Prove that the ideal <x> in Q[x] is maximal.

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Q: Prove that there are integers x and y such that 10x-13y=1

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Q: Prove that the ideal (x2 + 1) is prime in Z[x] but not maximal in Z[x].

A: To prove that the ideal <x2+1> is prime in Z[x] but not maximal in Z[x]. Use the definition,…

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: 26. Let R be an integral domain. Show that if the only ideals in R are {0} and R itself, R must be a…

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Q: с. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

A: (C). Prove that neither 2 nor 17 are prime elements in Zi.   Note : An integer a+ib in Zi is a prime…

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: (2) Prove that Q(V2) and Q(V3) are not isomorphic as fields.

A: The complete solutions are given below

Q: Define g : (-1, 1) → R by g(x) 1- x2 " Show that |(-1, 1)| = |R| by showing that g is a bijection

A: Concept Used: For a function f:A→B If f is a bijection, then |A|=|B| Consider the function g:(-1,…

Q: Show that U(8) is isomorphic to U(12).

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Q: Show that Z4 is not isomorphic to U(12) by comparing the orders of elements.

A: z4=0,1,2,3 is group w.r to +order of elementsoa=n⇒an=e where n is least positive…

Q: 4. Prove C(n, r) = C(n- 1,r- 1) +C(n- 1, r) %3D using a combinatorial proof.

A: We know, C(n,r) = n!/((n-r)!r!)

Q: Prove: |a| >= 0  for all a's in the reals.

A: It is known that the absolute value is the distance of the point or number from the origin.

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

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Q: Use Eisenstein criterion to verify that, if p =2, f(x) =x2-2 is iireducible over rational numbers,…

A: Eisenstein's criterion states that if Q(x)=anxn+......+a1x+a0, a polynomial with integer…

Q: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

Q: prove R is not homomorphic to R^2

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

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Q: Prove that the set 3Z + 2 = {3n + 2: n E Z} is denumerable. Hint: It may be easier to build a…

A: prove that set 3ℤ+2=3n+2: n∈ℤ is denumerable. defined a function ϕ:ℤ→3ℤ+2 by ϕn=3n+2 prove that it…

Q: Prove that 8x3 - 6x - 1 is irreducible over Q.

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Q: Prove

A: Let                 f(x)=x3+12x2+18x+6 By Eisenstein's Irreducible Criterion

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: 2) Show that g(x) = 215 x* + 623 x3 - 37 x2 + 111 x- 15 is irreducible over %3D Q.

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Q: 2. Prove that [0,1] and U-1 An , where A,= [2n, 2n+1] have the same cardinality. %3D

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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