Show that Z[√−D] is not a unique factorization domain, where D is squarefree and positive

Q: Determine whether T={(a, - a) | aEZ} is a subring of ZxZ.

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Q: For f(x), g(x), and Zn[x]in the given question ,, find the greatest common divisor d(x) of f(x) and…

A: Given: f(x)=x3+2x2+2 and g(x)=2x5+2x4+x2+2 in Z3[x]. To determine: the greatest common divisor d(x)…

Q: Show that MnXn and Rn^2 are isomorphic.

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Q: Show that Q(i) is a subfield of c

A: Subfield of a field

Q: (a) Consider the inner product on P2 defined by (p, q) = aobo + 2a1b1 + 3azb2 for each polynomial…

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Q: for the polynomil bellow 1 and 3 are zeros h(x)=x^4-2x^3-7x^2+14x-6 express h(x) as a product of…

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Q: show that Q has infinitely many distinct quadratic extensions in C

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Q: Prove that K = {x+yv3 ] x, y E Z} is a subring of R.

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Q: c) The operation (ao + a1r + azx², bo + bịx + b,x²) = aobo + a¡b1 – azbz defined on P2 has all the…

A: We have to check whether the operation defined on P2 satisfies all the properties to be an inner…

Q: et A be m x n and let  be in R^n. If A has a row of zeros, show that Ax has a zero entry

A: We have to solve given problem:

Q: Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].

A: We have to show that 2 is equal to the product of a unit and the square of an irreducible in Zi. The…

Q: Prove that Z3 x Zz is isomorphic to Zg.

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Q: Show that A = has two different factorizations of the form A = L · U where L is lower triangular…

A: Given: A=0001. To show: A has two different factorization of the form A=L·U, where L is lower…

Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

Q: In the field GF(p"), show that for every positive divisor d of n, - x has an irreducible factor over…

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Q: e of all polynomials in C[0, ed in C[0, 1] with respect to

A: To prove a subset W is a subspace of V, it must satisfy 3 conditions: 1) 0 ∊ W. 2) closed under…

Q: Find the standard inner product P2 of the given polynomials. on p = -3+ 3x + x², q = 2 + 5x – 4x?.…

A: According our guideline i solve only first.For second solution please resubmit next question again.

Q: Find the gcd of (x^4+x^3+x^2+2x+3) and (x^3+4x^2+2x+3) in Z5.

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Q: Describe the group of the polynomial (x4 – 5x2 + 6) ∊ ℚ[x] over ℚ.

A: Please see the next step for solution

Q: Show that the function (, -): R² × R? → R defined by setting = 1122+ 2x12 + 2y172 + 8y142 = is an…

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Q: Show that ZV-6 is not a UFD by factoring 10 in two ways. Why does this prove that Z[V-6] is not a…

A: Given, Z-6 is not a unique factor domain.To find:Z-6 is not a principal ideal domain.

Q: Use the Factor Theorem to prove that x + c is a factor ofxn + cn if n Ú 1 is an odd integer.

A: To prove the given statement.

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

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Q: Are the polynomials x? + 1,x? – 1, 2x – 1 linearly independent in P2?

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Q: The set of polynomials {1, x² + 3x“, 2x² – 7x4} is a linearly independent set in P4.

A: NOTE: Due to time limit answer of 1 is given here, for other please ask in a different question.…

Q: Find all (real) values of k for which A is diagonalizable

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Q: Is the polynomial x2 – 4 in the ideal generated by the polynomials x + 22 – 4x – 4, x – 22 – 4x + 4,…

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Q: Identify the splitting field of the polynomial x2 – aª over Q(7*)

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Q: show that Q[x]/(3x⁴+2x³+1) is a field. Here (3x⁴+2x³+1) is the principal ideal generated by a…

A: Given factor ring is ℚx3x4+2x3+1 where 3x4+2x3+1 is the principal ideal generated by the polynomial…

Q: Find (a) ATA and (b) AAT. Show that each of these products is symmetric.

A: a).

Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: 5. Suppose A = PDP¯1, where P is 2 × 2 and D = 9I. Let B = 51 – 3A+ A². Show that B is…

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Q: Let P3 have the inner product

A: Given, pt=t2 and qt=2t-3. given by evaluation -2,-1,0,1, and 2. p-2=4q-2=-4-3=-7 p-1=1q-1=-2-3=-5…

Q: Prove that Z[√ -2] and Z[√ 2] are unique factorization domains.

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Show that Z4 is not isomorphic to U(12) by comparing the orders of elements.

A: z4=0,1,2,3 is group w.r to +order of elementsoa=n⇒an=e where n is least positive…

Q: *** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

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Q: Prove that D24 and S4 are not isomorphic

A: Since D24 ={a,b: a 12 =b2 =e (identity) and ab=ba-1 } This implies that D24  has 12 reflection and…

Q: Find the minimal polynomial and the conjugates over Q of the algebraic V-3 number 5

A: Algebraic number :- An algebraic number is a number that is a root of a non-zero polynomial in one…

Q: By using Eisenstein's criterion we can conclude that the polynomials x^3+4x^2+3 and…

A: Using Eisenstein's criterion we can check a polynomial is reducible or irreducible. For that some…

Q: Prove the following statement: If x € Z, shares a common nontrivial factor with n, then a does not…

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Q: Find the standard inner product on P of the given polynomials. p= -2+ 4x + a2, q = 3+ 4x 3a2. (p, q)

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Q: Determine whether T={(a, - a) | aEZ} is a subring of zxZ

A: If (R,+,.) be a ring and S be a nonempty subset of R then S is a subring of R if and only if 1) a∈S…

Q: Prove that C is not the splitting field of any polynomial in Q[x].

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Q: Let P2(R) have the inner product, (p, q) = f* p(x) q(x) dx, Vp, q € P₂ (R). Find the best…

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Q: Is Z6 a subring of Z12?

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Q: Given two polynomials in from of x^3+x^2+x+c =0 x^3+x^2+x+c =0

A: Given: To find their intersection using matrices. Given two polynomials in from of…

Q: (a) Consider the inner product on P2 defined by (p, q) = aobo + 2a¡b1 + 3azb2 for each polynomial…

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Q: Determine whether T={(a, -a) | a EZ} is a subring of ZxZ-

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