a) Find a monic polynomial in (Z/8Z) [x] with more distinct roots (in Z/8Z) than its degree. b) Find a non-0 monic polynomial in (Z/8Z) [x] of degree 4 such that every a € Z/8Z is a root.

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A: In mathematics, a splitting field is a field extension that contains all of the roots of a given…

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A: Sol

Q: Exercise 47 (a) Show that there exists a unique polynomial po of degree < 2 such that po(-1) = 1,…

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Q: How to find the characteristic polynomial of T: P3(R) → P3(R) defined by T(p) = xp' – 4p

A: To find the characteristic polynomial for the given linear transformation.

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A: Let F be a field of characteristic 2 with more than two elements, and let x, y be arbitrary elements…

Q: Give an example of a degree three monic polynomial in Z[x] that is irreducible in Q[x] but whose…

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Q: Let & be a zero of f(x)=x2 +2x+2 in some extension field of Z3. Find the other zero of f (x) in…

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Q: The polynomial is irreducible over Z5. (Do NOT show this.) Let a be a zero of f(x) in some extension…

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Q: Let M = 0 4 -4 -5 9 -6 € Mat(3, R). -4 5 -2 1. What is the characteristic polynomial of M (give your…

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Q: Let α be a zero of f (x) = x2 + 2x + 2 in some extension field of Z3.Find the other zero of f (x) in…

A: Given: α is a zero of fx=x2+2x+2 in some extension field of Z3. We have to find the other zero of…

Q: 2. Determine the Jordan Canonical Form of A if the characteristic polynomial of A is given by…

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Q: Find all values of p such that Z„[x]/(x² + 1) is a field.

A: Given problem is :

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Q: Prove or disprove that there exist nonconstant polynomials in Zp[x] that are units if p is prime.

A: