Answer asap. Mark ALL of the functions from ℤ12 to ℤ12 that are invertible. The notation [x] is an abbreviation for [x]12, the congruence class of x mod 12.

Q: Find the dimension of the splitting field of each of polynomials x+1 and x² - 2.

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Q: Let G=Z5xZ6X Z9 X Z12- Find the elementary divisors and invariant factors of G.

A: Solution :-

Q: PERFORM THE OPERATOR AS SPECIFIED IN EACH ITEM. 1. 9E7 16 2.9876 101 X 100111 2 +11101112 4.675 8 X…

A: (1)  9 E 7 16= 9×162 + 14×16 + 7×1 = 253510      1001112 = 1×25 + 0 + 0 + 1×22 + 1×21 +1= 3910…

Q: The multiplicative inverse of 1+3x in Z,[x] is: О бх+1 O 1/(3x+1) None

A: Inverse property: a*a-1=e In Polynomials    A non-constant polynomial can never have an inverse…

Q: Find a zero divisor, if possible, in each of the following. If not possible, explain y not. (a)…

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Q: a. Give the smallest extension of Q that contains all the zeros of x4 - 2x2 – 15.

A: We shall answer first question only as you have asked diff question without mentioning. For others…

Q: 1. Find irr(a, Q) and deg(a, Q) for the given algebraic number a = √2+ i C. [Hint: Find irr(a, Q) as…

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Q: = a O a = a for all positive integers a. = a O b = b O a for all positive integers a and b. = If a >…

A: We apply the given formulas repeatedly.

Q: A car costs x wholesale. The price the dealer pays is given by the function s(x) = = x-1 – 200 where…

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Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: The fields Q(√3) and Q(11) are isomorphic. O True O False

A: We have to determine that whether the following statement is true or false ; The fields Q33 and Q113…

Q: Construct the multiplication table for Zs. Using this table, or otherwise, determine the following x…

A: Solution

Q: Enter the multiplicative inverse of the element [21]49 X+ [18]49 in the ring Z49[x].

A: Extended Euclidean Algorithm

Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

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Q: PLEASE TYPE ONLY*** Exercise 4.2.3: Find a counterexample. Find a counterexample to show that each…

A: we have to write the counterexample to show that the following statements are false

Q: Construct the addition table for Zs. Using this table, or otherwise, determine the following x in…

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Q: Exercise 4.6. Prove that if h1, h2 are two greatest common divisors of f1, ..., fs, then there…

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Q: Is Q [x]/⟨x2 -5x + 6⟩ a field? Why?

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Q: Help me with my computational theory homework

A: Approach to solving the question with detailed explanation: Regular expression for the complement of…

Q: Find the following Wronskian: w(2, 31, —е 4, te 4) O 8 O 1536e8! O -8 O - 1536e8: O 1536e¬8 -8r O -…

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Q: The multiplicative inverse of 1+ 3x in Z,[x] is: None 1/(3x+1) O 4x+1 О бх+1

A: Use the property of ring of polynomials to solve this problem.

Q: Let a be a non-zero element of Z43. w| 42. Show that [3] is a primitive root of Z43 and that [2] is…

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Q: 2. Find the following elements in Z₁9: a. 13 X19 17 b. 13 +19 17 c. -12 (the additive inverse of…

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Q: Find the inverse of 1 + √2 in Z[√2]. What is the multiplicativeorder of 1 + √2?

A: Here ℤ2=a+b2/a, b∈ℤ We need to find multiplicative inverse of 1+2 in  ℤ2. An element b∈R is said to…

Q: Let F be an ordered field and re F. Show that if x 0, then x² > 0.

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Q: solve this question: Let J5 = {0, 1, 2, 3, 4}, and define functions f : J5→J5 and g : J5→J5 as…

A: S we can clearly see that f(0) = g(0), f(1) = g(1), f(2) = g(2) but f(3) = g(3) & f(4)…

Q: Find the characteristic equation of the recursive relation an 송,Mn 2 1. O 3r – 1=0 Or+3=0 Or-3= 0…

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Q: Find the following Wronskian: W(2, 5t, -e-2¹, te-2¹) O-160e-4t 04 00 O-160e4 O 160et O-4 O 160e-4t

A: We want to find the given Wronskian

Q: Find the inverse of 6 − 5√7 in the field ℚ[ √7]

A: To find the inverse of 6 − 5√7 in the field ℚ[ √7].

Q: Each of the following is incorrect. Explain why or give a counterexample. 1. There are 60 zero…

A: Use the formula of zero divisors in Zm and the properties of Euler's Phi function as in next step.

Q: is the columd of matnla A Explain why Oineanly nメn an A is invertible. independent when are 1 6

A: To explain why the columns of an n x n matrix A are linearly independent when A is invertible.

Q: Determine whether e(x)=15x+4(mod 26) and e(x)= 4x+5(mod 26) are invertibles or not. If yes, find the…

A: We are asked to determine if the given equation are invertible and if they are we have to find the…

Q: Prove that if f(n) is multiplicative, then so is f(n)/n.

A: We have to prove that if f(n) is multiplicative, then f(n)n is also a multiplicative function.