19.2. Rings, Fields and Polynomials

Try these first

1. 1.

   What is the main property that distinguishes a field from a ring?

   Hint

   Think about multiplicative inverses.

2. 2.

   List the two operations defined over a ring.

   Hint

   These are commonly notated as '+' and '×'.

3. 3.

   What must a set satisfy to be called a ring?

   • Only multiplication axioms
   • Only addition axioms
   • Both addition and multiplication axioms

   Hint

   Think about the characteristics of each operation.

4. 4.

   In a field, is it true that every non-zero element has an inverse?

   • True
   • False

   Hint

   Reflect on what defines a field.

5. 5.

   Prove that the set of integers under addition and multiplication modulo N forms a ring.

   Hint

   Consider each operation carefully within the modulus structure.

6. 6.

   Using the polynomial a(x) = 2x + 1 in Z_3, compute a(2) and interpret the meaning of this result.

   Hint

   Remember the significance of assigning values to polynomials.