9. Let R[[X]] be the ring of formal power series with coefficients in R. Prove: (i) f = ao +... is a unit of R[[X]] if, and only if, ao is a unit of R. (ii) If f is nilpotent, then an is nilpotent for every n € N. (iii) Let K be a field. Prove that the non-zero ideals of K [[X]] are of the form (X¹).

Q: Let F be a field and f(x) E F[x] be of degree

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Q: Let F be a field. Show that if det(A)=0 with A E Mn(F) then A is a zero-divisor

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Q: .4. If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F…

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Q: Show that f(x) = x² + 1 is irreduci. be a zero of f(x) in an extension ist nine elements. 2₂ (x).…

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Q: The number of elements in a finite field of characteristic n p is of the form p ", where n being…

A: We have to show that the number of elements in a finite field of characteristic p is of the form pn…

Q: (a) Let m and n be integers greater than 1. What is the order of the element (1,1) in Z/mZ x Z/nZ?…

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Q: Consider r¹7 in Q[r]. (a) Find the splitting field E of over Q. (b) Compute [E : Q].

A: Given: x7-7 in ℚxTo find: a) Splitting field E of  x7-7 in ℚx              b) E:Q

Q: Let U = (0, 1) U (2, 3) U (4, 5) be a union of three open [0, 10] \ U. Show that A is compact. 2.…

A: Que  Ans : given         U = (0,1)U (2,3) U (4,5)  be unian              of 3 open sets    A= [0,10]…

Q: Let K be an extension of a field F and let f(x) € F[x] be a polynomial of degree n≥2. Then a€K is a…

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Q: Assume C and D are chosen at random from {1, 2 . . . , n}. Let pn be the probability that C + D is a…

A: Given: C and D are chosen at random from {1, 2 . . . , n}. Then the number of pairs (c, d) with sum…

Q: Let T be the set of all sequences {an} of elements of Z.  Prove the following. (i) T is an integral…

A: A commutative ring with unity (i.e., one multiplicative identity) and no zero divisors is referred…

Q: Let f: C C defined by f(z) = 1 (z-2)(z-1)³ Express "f" as a Laurent series which is valid for the…

A: Laurent series expansion represents an power series with positive and negative terms as well.  So,…

Q: Let F/K be a field extension and a, b E F be algebraic over K. If a has degree m over K and b 0 has…

A: The data given in this question are: F/K is a field extension. a and b are elements of F. a has a…

Q: Define an algebraically closed field. Show that field E is algebraically closed if and only if every…

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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: Use mathematical induction to show that (4k – 1) = n(2n + 1) k=1 for all n > 1. 2. Let U = (0, 1) U…

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Q: 14. Prove that F Z[]/(1+ 2i) is a field. How many elements are in F? What is the characteristic of…

A: Z[I]\(1+2i) and the number of elements

Q: Use the definition to show that 4n – 5 lim 2n + 3 - 2. n00 3.

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Q: 8. Show that cos a – cos y| < |x – y| for all æ and y.

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Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Use mathematical induction to show that Σ (4k – 1) = n(2n + 1) %3D k=1

A: To prove the given statement by using mathematical induction.

Q: Auume we know that m(X) = X4 + X + 1 ∈ Z2[X] is irreducible. Let K denote the field Z2[X]/(m(X)) and…

A: a) The identity is (b1+b2+...+bk)2=b12+b22+...+bk2(b1​+b2​+...+bk​)2=b12​+b22​+...+bk2​.  b) The…

Q: 5. For n,k € Z such that 0 ≤ k ≤n, define (3). Note that, by convention, 0! = 1. (a) Prove that () =…

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Q: 2. Define the function a(n) as follows: n = 1 n = 2 2a(n-1) + a(n − 2) n ≥ 3 5 a(n) = 10 and let…

A: Given that a1=5, a2=10, an=2an-1+an-2, for n≥3 Our aim is to prove that an&lt;3n for n≥3 using the…

Q: 1. Let q-p" be a prime power. Show that X+ 1 is irreducible in F [X] if and only if q=3 mod4. (Hint:…

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Q: Use the definition to show that x2 + x -6 lim = 5. x - 2 4.

A: I have just used the definition of limit

Q: Let h : N x N → N be the function h(x, y) = 3x + 2y. Use strong induction to prove that if n EN and…

A: Given that h : ℕ×ℕ→ℕ be the function h(x,y)=3x+2y. Then we have to prove by using strong induction…

Q: 3. Suppose p is prime and E is a field extension of Zp. Suppose there is a € E which is a zero of rP…

A: Given: p is prime and E is field extension of  Zp. α∈ E  is a root of xp -x+1. (b): To prove: mα,zpx…

Q: Let n > 1 be an integer. We say that [x] ∈ℤ/n is a zero divisor if there exists [y] ∈ ℤ/n − {[0]}…

A: Let n&gt;1 be an integer.  Given details, [x]∈ℤ/n is a zero divisor if there exists [y]∈ℤ/n-{[0]}…

Q: a. Suppose p is an odd prime, and let A := a, b € ? (a) Suppose there are ao, bo € Z such that p =…

A: Given p is an odd prime and  A=ab-ba:a,b∈ℤp Suppose, there are a0,b0∈ℤ such that p=a02+b02. Then, by…

Q: Construct the finite field F under addition and multiplication modulo 3 whose elements are…

A: Given: Consider the given F to be the finite field under addition and multiplication mod 3 hose…

Q: characteristic of Z[i]/(2 + i). nhoracteristic of Z[i]/(c

A: Given,            Zi2+i

Q: 1. Let F be the field with 2 elements. The number of (ordered) bases of F" over F is (а) п (b) п?…

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Q: 1. Which of the following is true.? Z₂[x]/(x³ + x + 1) = Z₂[x]/(x³ + x² + x + 1) C[x]/(x² + 1) Is a…

A: Given:  1) a) ℤ2x/x3+x+1≅ℤ2x/x3+x2+x+1 b) ℂ[x]/x2+1 is a field. c) x2+100x+2500-n2 is irreducible…

Q: X[k] =a* [u[k] - u[k − N]]

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Q: (2xn+7)/4 for n = 1, 2, 3,. Show that {Tn} is convergent and find its limit. 7. Define a sequence…

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Q: Let F be a field and let f(x), g(x), n(x). k(x), and h(x) be polynomials in F[x]. Prove that if f(x)…

A: Given thatLet F be a field and let f(x) , g(x) , n(x) , k(x) , and h(x) be polynomials in F[x] .We…

Q: Let F be a field. Two variations of the ring of polynomial over F are the ring of formal power…

A: To show that F[[X]] (the ring of formal power series) and F((X)) (the ring of formal Laurent series)…

Q: 9. Let F be a field of characteristic 2 with at least 2 elements. Then for some x, y in F (x + y)³ ‡…

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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.