Let A = Z[r]deg

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

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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: Give the definition of the following Concepts: 1- Least Upper bound Property (LUBP) 2- Upper Riemann…

A: Bounded Above: A nonempty subset S of ℝ is said to be bounded above if there exists an element u∈ℝ…

Q: Prove that the piece-wise function defined by 1 3x if r -2 is integrable on [-4, 7]. (Hint: do not…

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

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Q: Prove by structural induction that F C F2

A: Given that, F1 such that, 3∈F1, n∈F1⇒n2∈F1. and F2 such that, 3∈F2, n,m∈F2⇒nm∈F2. Consider Pn be…

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

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Q: Find example where of is continuous and an - bijective but f¹ is not

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Q: Let a ≠ b in a field F. Show that x + a and x + b are relatively prime in F[x].

A: Definition of relatively prime: A polynomial in fx and gx in Fx is said to relatively prime if the…

Q: Prove that fVf · dr = 0.

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Q: Assume that the coefficients in the expansion f(x) = anx" n=0 satisfy Enanx" – E Anx"+1 = 0. n=1 n=0…

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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: Define a mapping fn : Z → Z by fn(x) = 1 – nx for each integer n E Z. a. For which values of n is fn…

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Q: (3) Prove that the piece-wise function defined by 1 3x if x -2 is integrable on [-4, 7]. (Hint: do…

A: The given question is related with integrability. Given a piece -wise function…

Q: In Exercises 67-72, express the area under the graph as a limit using theapproximation indicated (in…

A: fx=x-1, 1, 7 Let f be continuous and nonnegative on the interval a, b. Then the area A of the region…

Q: Let [a] ϵ Zn- Prove ([a]) = ([n - a]).

A: Given, a∈Zn- The objective is to show that, a=n-a. Let x∈a. Then, x=ra,r∈Zn=ra=-rn-a=-rn+ra as…

Q: The number of reducible monic polynomials of degree 2 over Zz is

A: The number of reducible monic polynomials of degree 2 over Z3 is

Q: are two bounded and integrable fiunctions on [a, b] and there exists a number >0 such that | f%(x) |…

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Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

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

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Q: Show that the Modulus Function            f : R → R, given by f(x) = | x |, is neither one-one nor…

A: Consider, f: R → R given by f(x) = | x |Where,| x | is x, if x is positive or 0| x | is – x, if x is…

Q: .4. consider the funchon f(x)= Vox-18. for the domain [3,00). Find (x), Where f.-! is the Įnverse of…

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Q: Let N be a nonempty set and B a o-ring on N. Let µ, v be two measures on B and μ(A) < ν(Α), A € B.…

A: Let Ω be a non- empty set. And B a sigma ring on Ω. Let μ,ν be the two measures on B And ΩA≤νA ,…

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

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Q: Let f(x) = Vsin x. Compute the midpoint sum for f corresponding to the partition of [0, 2] into 20…

A: Draw the graph of the function .

Q: . Let X;, i = 1,2, · . be a sequence of random variables defined on (N, F). Let (N, G) be the…

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Q: 1 Use the Cauchy's integral test to show that Σ diverges n. In n. In In n n=2

A: Cauchy -Integral  Test series ∑n=1∞ an      converges if integral ∫1∞ f(x) dx finite and  diverge if…

Q: Let n EZ. Show that f(x)= x³ – nx +2 is irreducible over Q for n+-1,3,5. 72

A: Given: fx=x3-nx+2, n∈ℤ. To show fx is irreducible over ℚ for n≠-1,3,5.

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: Showthat S subring of RIKRZ+R3

A: For the solution of the problem follow the next step.

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

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Q: Show that f(x)= x14 – 6x + 75 is irreducible over Q.

A: Eisenstein's Irreducibility Criterion: Let us consider polynomial…

Q: p be an irreducible reprsentation of G

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Q: use interediate value theorem o prove that 2cosk -3x=O a- real zero to between O and I

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Let f be a bounded function on I = [a, b]. Also, suppose P and Q are distinct partitions of I. It is…

A: Riemann sum over two different partitions P and Q of a given interval may or may not be same. So it…

Q: be an integral domain of characteristic 2. Show that va,bED, (a+ b)4 = aª + bª. Let D

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Q: Consider the f₁ (x) = funtion if 0<x< 1/ if x = 0 oraz in 5 Show that fn is integrable by using the…

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Q: A subring of an integral domain is also an integral domain. Select one: O True O False

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Q: Let D be an integral domain with four elements, D = {O, e, a, b }, where e is the unity.   Prove…

A: Let D be an integral domain with four elements, D=o,e,a,b, where e is the unity. The objective is to…

Q: (c) Prove that 10 is not separable.

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Q: Use the Integral Test to determine the convergence or divergence of theseries

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Q: A subset of Q that contains its supremum but not its infimum.

A: Supremum of Set :- Let S be set of real number which is bounded above. Then M ∈ R is said to be…

Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

Q: Find all monic irreducible polynomials of degree 2 over Z3.

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Q: Let the chor a cteristic polynomial A, ムA= Ala)(入+). Then det (A) =?

A: The characteristic polynomial of matrix A = λ(λ2+1)(λ+1)

Q: Find infinitely many polynomials f(x) in Z3[x] such that f(a) = 0 forall a in Z3.

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Q: 3. Prove that the sandwitch theorem.

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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

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.