7. Let a be a root of X² - bX+c, for b, c integers such that b² - 4c < 0. Supposethat unique factorisation holds in Z[a]. Let m, n be relatively prime positive integerssuch that mn is expressible as x² + bxy + cy² for integers x and y. Show that bothm and n are expressible in this form.

Answered: Image /qna-images/answer/fe91a1f4-681b-429c-857e-5ce475bab768.jpg

Answered: 7. Let a be a root of X² - bX+c, for b,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

1) Are the polynomials x³+2x +2 and x4 + 3x -2 equal to mod(x² + 2x + 2) in Q[x]?
3. Show that Z10 is isomorphic to Z2 x Z5, but that Zg is not isomorphic to Z2 × Z4.
Solve and answer the following problems as indicated.2) Create the multiplication table for {0,3,6,9,12} under multiplication modulo 15. This is a subring of ℤ_15. Determine its unity.3) Find all the roots of f(x)=x^2+3x+2 in ℤ_6. List them in increasing order, separated by commas and without spaces.
1. Compute the resultant of x polynomials have a common factor in Q[x]? Explain your reasoning. - 3x* – 2x + 3x² + 7x + 6 and x +x² + 1. Do these
Find the minimal polynomial and the conjugates over Q of the algebraic number V-3
Showthat S subring of RIKRZ +R;
5. Suppose that GF (2³) is defined by the irreducible polynomial x³ + x + 1. From 4., deduce that the multiplicative inverse 101-¹ of 101 in GF(2³) is: [Hint: The elements of GF (2³) can be represented by polynomials ax² + bx+c whose coefficients a, b, c are in GF (2), which in this case are the bits 0,1. We can represent the polynomials by the corresponding bit-strings: e.g., x² + 1 → 101 and x + 1 → 011. Then if we reduce he product of the polynomials modulo the irreducible polynomial x³ + x + 1, this will correspond to the product: 010 · 101. What was the product of the polynomials congruent to? The answer should follow.]
The set of all polynomials of the form at³ + bt² + ct + d + 1 where, a, b, c, d are any real numbers, is not closed under addition but is closed under scalar multiplication. Select one: True False
Explain why 3Z is not an ideal in Q
Let a, b and c a=1b=0c = 9 example with enrolment number 001054321, you would take a = 0, b = 5 and c = = 4) and then take each of them (mod 2). So in the example above, a, b and c would become 0, 1, 0 respectively (each digit taken (mod 2)). Consider the polynomial x5+ax3 + bx² + cx + 1. (a) Create a polynomial code of codeword length 8 from this generating polynomial. (b) Is your code cyclic? Justify your answer.
2. The volume of a tissue box is represented by the polynomial 22-32² +42-9. If the height of the tissue box is z+2, what is the area of the base of the tissue box? O 2x³+4x²+5x+14+1/2 2z³+z+2-2 2z²³-42² +52-6+ O22²³ +7z+18+ PREVIOUS 2 of
3) Let the order on the X=(-00₂ √₂)NQ be the known order. Find the supremum Supremum and infimum of the set A= (-√2-√2) NQ

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7. Let a be a root of X² - bX+c, for b, c integers such that b² - 4c < 0. Suppose that unique factorisation holds in Z[a]. Let m, n be relatively prime positive integers such that mn is expressible as x² + bxy + cy² for integers x and y. Show that both m and n are expressible in this form.