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Evaluation and Properties of Polynomials and Fields in Abstract Algebra

This document explores various computations and definitions in abstract algebra, specifically focusing on polynomial rings, fields, and their properties. It covers evaluation homomorphisms, the count of elements in rings, factorization of polynomials, and identification of prime numbers. By answering true/false questions on concepts such as zero divisors, field quotients, and irreducibility, it aims to clarify foundational algebraic principles. Corrections are provided where initial statements may have been incorrect, supporting deeper understanding of algebra's structure.

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Evaluation and Properties of Polynomials and Fields in Abstract Algebra

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  1. Computation Definition True/False 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50

  2. Computation– 10 Points • QUESTION: • Compute the following evaluation homomorphism: • ANSWER:

  3. Computation– 20 Points • QUESTION: • How many elements are there in the ring • ANSWER: • , , , • There are 4 elements

  4. Computation– 30 Points • QUESTION: • Find all zeros: • ANSWER: • 4 and 0

  5. Computation– 40 Points • QUESTION: • Factor in • ANSWER:

  6. Computation– 50 Points • QUESTION: • Find all prime numbers p such that x+2 is a factor of , where p is a prime. • ANSWER: • A zero is -2 • p is the factors of 15 • p=3, 5

  7. True/False– 10 Points • QUESTION: • The fact that D has no divisors of 0 was used strongly several times in the construction of a field F of quotients of the integral domain D. • ANSWER: • False

  8. True/False– 20 Points • QUESTION: • has no zeros in Q, but does have zeroes in R. • ANSWER: • True

  9. True/False– 30 Points • QUESTION: • If D is a field, then any field of quotients of D is isomorphic to D. • ANSWER: • True, D is its own field of quotients

  10. True/False– 40 Points • QUESTION: • If F is a field, all we can say is F[x] is an integral domain. • ANSWER: • True, the element is NOT a unit, therefore it has no multiplicative inverses (can’t call it a field).

  11. True/False– 50 Points • QUESTION: • The polynomial: is reducible over Q. • ANSWER: • False, using is irreducible for any prime p. (p = 5)

  12. Definition– 10 Points • QUESTION: • Verify the following the statement, if a correction is needed state the correction: • The field of quotients F is minimal. If L is any other field containing D, then there exists a map that is an isomorphism of F with a subfield of L. • ANSWER: • This statement is correct

  13. Definition– 20 Points • QUESTION: • Verify the following the statement, if a correction is needed state the correction: • R[x], the set of all polynomials with coefficients from R, is a ring if R is not commutative, so then is R[x]. If R has unity 1≠0, then 1 is also unity for R[x]. • ANSWER: • This statement is incorrect: • R[x] the set of all polynomials with coefficients from R is a ring if R is not commutative, so then is R[x]. If R has unity 1≠0, then 1 is also unity for R[x].

  14. Definition– 40 Points • QUESTION: • Verify the following the statement, if a correction is needed state the correction: • Let with degree 2 or p, where p is a prime. Then is reducible iff it has a zero. • ANSWER: • The statement is incorrect:

  15. Definition– 50 Points • QUESTION: • Verify the following the statement, if a correction is needed state the correction: • A polynomial is irreducible over the field F if and only if for any polynomials • ANSWER: • The statement is incorrect. We must require that g(x) and h(x) have degree less than the degree of f(x), and that the polynomial is nonconstant. • A nonconstantpolynomial is irreducible over the field F if and only if for any polynomials both of degree less than the degree of .

  16. DAILY DOUBLE

  17. DAILY DOUBLE • QUESTION: • Find a polynomial of degree > 0 in that is a unit. • ANSWER: • therefore is a unit with a degree of 1

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