9. Show that if a is a root of X²-bX+c with b, c integers and 6² - 4c < 0, then there are only finitely many units in Z[a].

Q: (c) 2, 3, 4+ V10 and 4. V10 are irreducible elements of R.

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Q: x - x3 + x² – x + 1 is reducible over Q True False

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Q: Let a and b be integers, at least one of them not 0. Prove that an integer e can be expressed as a…

A: Let a and b be intgers ateast one of them not 0. Our Aim is to prove that an integer e can be…

Q: Let a and b be integers with GCD(a, b) = d and let a = da', b = db'. Prove that GCD(a', b')=1

A: and  be integer such that .It is given that  and .We have to prove .Note: According to the…

Q: . Show that x2 + 3 and x2 + x + 1 over Q have same splitting field.

A:

Q: Suppose that there exist integers x and y such that ax + by = 1. Show that a and b are relatively…

A:

Q: 1. Prove or disprove: The following gives an inner product on R².

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Q: Prove that x4 + 15x3 + 7 is irreducible over Q

A: In this question,  we have to prove that  x4 + 15x3 + 7 is irreducible over Q.

Q: Prove that Q is not isomorphic to Z.

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Q: left cosets

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Q: Suppose that a and b belong to an integral domain and b ≠ 0. Showthat &lt;ab&gt; is a proper subset…

A: Let R be an integral domain. A commutative ring R is said to be an integral domain if non zero…

Q: 5. Let r,ne N. Show that the product (n+1)(n+2)...(n+r) of r consecutive positive integers is…

A:

Q: 3. Let R-Z18. Determine the units and zero divisors of R giving the explicit unit or zero divisor…

A: We know that  no of units in ring Zn = phi(n), and  no of zero divisor of Zn = n - phi(n) -1

Q: n 5. Let n be a fixed positive integer. Denote by Po the set of all real polynomials of degree n or…

A: A nonempty set V together with two operations - vector addition and scalar multiplication is called…

Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

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Q: Show that for every positive integer n there exists an irreducible polynomial of degree n over Fp.

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Q: EXERCISES 4.1 Which elements of (Z₁0, ). are contained in ?

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Q: 7. Find a pair of integers (x,y) such that 4021x + 2014y= 1.

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Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: Determine the numbers a such that x a (p) is solvable for p = 7, 11, and 13. %3D

A: Given, x3≡amod p At p=7 We know that suppose p is rational prime p≡1mod 3 then x3≡a mod p is…

Q: Show that 1+3/ -5 is irreducible which is not prime in ZlV-5]

A: We know that ,  ℤ-5 = a + b-5 / a , b ∈ ℤ We need to show that , 1 + 3-5  is irreducible but not…

Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

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Q: 5. Find LU Factorizatioon of A = [5 2 1 1 3 2 7 1 2 3 2 1- Show the work!

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Q: Exercise 6.4. We say that a number î divides another number z if there exists an integer k such that…

A: We say that a number x divides another number z if there exists an integer k such that xk = z. Prove…

Q: Prove this statement: we have numbers {a1, a2, a3 .,a2n} which are even and positive, but 0 < {a1,…

A: As, we know that, A.M.≥G.M. ---(1) A.M. of numbers, a1,a2,a3...a2n=a1+a2+a3+a4...+a2nn G.M. of…

Q: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

A:

Q: 3. Show that Z10 is isomorphic to Z2 x Z5, but that Zg is not isomorphic to Z2 × Z4.

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Q: 1. Compute the resultant of x – 3x* – 2r + 3x + 7x + 6 and x* +² + 1. Do polynomials have a common…

A: Given: To calculate the resultant of .

Q: 3. Prove that if m is a positive integer and x is a real number, then 1 2 m 1 [mx] = [x] + \x + = |…

A: The greatest integer function is one of the most important real valued number theoretic function.…

Q: (6) Suppose A CR is countable. Show that there exists a real number r such that An(r+ A) = 0. (Here,…

A: We prove this by contradiction