Q2: Let Z be the set of integer numbers and , defined as follows Va, be Z, then ab = a + b+1 aob = a.b Show that whether (Z)is ring or not

Q: for any integer a, show that gcd(2a+1,9a+4)=1.

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Q: Let R[r] be the set of all expressions 00 a = do+ajx+azx += i=0 where a¡ E R for all nonnegative…

A: We will be solving part (b) as mentioned. (b). Given ℝx be set of all expressions…

Q: Prove: Let a > 0 and b > 0 then either a > 0 and b > 0 or a < 0 and b < 0. where a…

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Q: If R is an integer domain, show that it contains exactly two elements a that satisfy a? = a.

A: Given R is an integral domain. We have to show that it contains exactly two elements a that satisfy…

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

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: Prove that a b + 2c + 3d + c+ 2d + 3a Let a, b, c, d be positive real numbers. C d + 2a + 3b + d a +…

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Q: which is necessarily true When 121=7 ? A. 2 IS pure real c. 2 is positine B.Zis pure imaginary D.7is…

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Q: Factor each of the following a) x-xー4a b)xペー64

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Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

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Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: Let a be a nonzero real number. Prove that aman = am+n for all integers m, n > 0.

A: Let a be a non zero real number , where m and n are integers…

Q: Determine all ring elements that are both nilpotent elements andidempotents

A: Let R be a ring with identity element 1. We want to find all elements of R that are both nilpotent…

Q: stion T1_Q3 rove by contradiction that there are no positive integers p and q such that 4p²-q² = 25

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Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: Find the following commutators by applying the operators to an arbitrary function f(x) [x − y (x² +…

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Q: If R is the set of all real numbers, what do the cartesian products R × Rand R × R × R represent?

A: Cartesian product R × R represents all the coordinates in R2  For example (1,2) , (5,6) coordinates…

Q: Use a direct proof technique to prove the following theorems: For all integers r and y, r + y - 3 is…

A: We have to prove that , For all integers x and y , x2+y2-3 is not divisible by 4. We will have 3…

Q: x + y)(z + w) = xz + xw + yz + yw A.) Explain in your own words what this problem is asking. B.)…

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Q: Let be the following set of rational numbers: Consider the algebraic structure (R2, +, ·), where +…

A: Field :-  A commutative ring F with unity is called field, if each nonzero elements of F has…

Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

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Q: Use proof by contradiction to show that no integers  y and z exist for which 24y + 12z = 1.

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Q: Let a, b, d E Z. Prove using only the definition of divisibility (that is, no other lemmas or…

A: Solution:- It is given that ,    a , b , d ∈ℤ then , we have to prove that  If     d /  a   and  d /…

Q: Show that i e Q (/2 ). .

A: Show that i∉ℚ2.

Q: (a) Demonstrate that a polynomial is completely determined by its coefficients. That is, if n and m…

A: (d) Given that m1,m2,b1,b2 are real no. we need to express lm1,b1.lm2,b2 in quadratic form. We'll…

Q: For an integer t, prove that gcd(2t + 1, 9t + 4) = 1.

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Q: Show that ACB implies |A|≤|B|.

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Q: Let a and b be two positive real numbers. Let p > 1 and q> 1 be such that + = 1 (q is called the…

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Q: Prove that the set of numbers Q(√2) = { a+b√2: a, b = Q} is a field.

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Q: 4 Write the allIV Algebraic Identities with complete explanation.

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Q: et ZZ be the set of all integers. Let R = {(a, b): a + b is divisible by 3}. a. Show that R is not…

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Q: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

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