Two operations, stricter clubs

Mathematics · Abstract Algebra · ages 21-22

Name ______________________ Date ____________

In integers mod six, two times three is zero mod six. What is two times three as an ordinary integer?
Answer: ______________
Which of these structures is a field?
- The rational numbers
- The integers
- Two by two real matrices
- The even integers
Which of these structures is a field?
- Even integers
- Integers
- Rationals
The integers modulo 5 form a field, while the integers modulo 6 do not.
Circle one: True False
Integers mod seven form a field. How many nonzero elements does it have?
Answer: ______________
Mod 7, every nonzero hour inverts. What does that make mod 7?
- A field
- A ring with zero divisors
- Not even a ring
Which structure is an integral domain but NOT a field?
- The rational numbers
- Integers mod six
- The integers
- Two by two real matrices
Every field is an integral domain. True or false?
Circle one: True False
A student claims mod 6 is a field because it is a ring. What is the error?
- Rings are never fields
- Being a ring is not enough; zero divisors block inverses
- Fields forbid addition
Why do integers mod p form a field exactly when p is prime?
- Primes are always odd numbers
- Composite moduli are too large to be fields
- Integers mod p are always fields
- Prime p shares no factors with smaller positives so every nonzero element inverts, while composite moduli give zero divisors

Answer key

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

6 · Two times three is six, which wraps to zero mod six.
The rational numbers · Nonzero rationals all invert. Integers lack most inverses, matrices do not commute or invert, evens lack an identity.
Rationals · Every nonzero rational inverts, while integers lack most inverses and evens lack an identity.
True · Five is prime so every nonzero element inverts; six is composite so zero divisors block division.
6 · Seven total elements minus zero leaves six.
A field · Full inverses for nonzero elements promote the ring to a field.
The integers · Integers cancel but two has no integer inverse. The rest fail domain or are already fields.
True · True. Inverses let you cancel any nonzero factor, so zero divisors cannot exist.
Being a ring is not enough; zero divisors block inverses · Fields need division by nonzero elements, which zero divisors destroy.
Prime p shares no factors with smaller positives so every nonzero element inverts, while composite moduli give zero divisors · Bezout inverts numbers coprime to the modulus. Composites supply factor pairs that multiply to zero.