Rings, Fields and Their First Properties · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
به انگلیسی

Two operations, stricter conditions

Mathematics · Abstract Algebra · ages 21-22
Name ______________________   Date ____________

  1. The nonzero residues 3 and 4 have product 0 in Z12. What is their product as an ordinary integer?

    Answer: ______________

  2. Which ring requirement makes subtraction a - b possible for every pair of elements?

    • Additive inverses exist
    • Every nonzero element has a reciprocal
    • Multiplication of all elements commutes
    • Every nonzero product is positive

  3. With ordinary addition and multiplication, which structure is a field?

    • Even integers
    • Integers
    • Rationals

  4. Z5 is a field, while Z6 is not.

    Circle one:   True   False

  5. How many nonzero elements does the field Z7 have?

    Answer: ______________

  6. Z7 is a commutative ring with 1 not equal to 0, and every nonzero residue has an inverse. What is it?

    • A field
    • A ring with zero divisors
    • Not even a ring

  7. Which structure, with its usual operations, is an integral domain but not a field?

    • The rational field Q
    • The residue ring Z4
    • The polynomial ring Q[x]
    • The one-element zero ring

  8. Every field is an integral domain.

    Circle one:   True   False

  9. A student claims Z6 is a field because it is a ring. What is the error?

    • Every ring lacks a multiplicative identity, including fields
    • A ring need not have inverses for nonzero elements
    • A field permits multiplication but has no addition operation

  10. Integers s and t satisfy sa + tn = 1. Which expression gives a multiplicative inverse of a modulo positive n?

    • t modulo n
    • n modulo n
    • a modulo n
    • s modulo n
LightMySky · lightmysky.comW1-mt_BmhIook13o-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Two operations, stricter conditions W1-mt_BmhIook13o-s1

  1. 12 · The ordinary product is 12, which reduces to 0 modulo 12.
  2. Additive inverses exist · Subtraction is addition of the additive inverse: a - b = a + (-b).
  3. Rationals · Every nonzero rational a/b has the rational inverse b/a.
  4. True · Five is prime. Six is composite and has nonzero zero divisors.
  5. 6 · Its seven residues include one zero residue, leaving six nonzero elements.
  6. A field · The stated conditions are precisely the field conditions, and 7 is prime.
  7. The polynomial ring Q[x] · Q[x] has no zero divisors, but the nonconstant polynomial x has no polynomial inverse.
  8. True · A field is nonzero and commutative, and inverse multiplication rules out nonzero zero divisors.
  9. A ring need not have inverses for nonzero elements · A ring alone need not be a field. In Z6, residue 2 has no inverse.
  10. s modulo n · Reducing the identity modulo n removes tn and leaves sa = 1.
Worksheet · LightMySky