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Computation

Computation. Definition. True/False. 10. 10. 10. 20. 20. 20. 30. 30. 30. 40. 40. 40. 50. 50. 5 0. Computation– 10 Points. QUESTION: Compute the following evaluation homomorphism: ANSWER:. Computation– 20 Points. QUESTION: How many elements are there in the ring ANSWER:

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Computation

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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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