19.2.7. Definition of a Field

Try these first

1. 1.

   What is a field?

   Hint

   Think about mathematical structures you have encountered.

2. 2.

   List the three main axioms for a field.

   Hint

   These involve operations and properties.

3. 3.

   What is required for a set to be a field?

   • They must include negative numbers
   • It must form an Abelian group under addition
   • It must contain at least one zero

   Hint

   Consider the definitions of mathematical structures.

4. 4.

   In a field, if x ∙ y = 0, what can we conclude?

   • True
   • False

   Hint

   Review the properties of fields.

5. 5.

   Prove that the set of rational numbers Q forms a field.

   Hint

   Reference the axioms discussed.

6. 6.

   Given a finite field constructed using Z_5, show the elements and their multiplication table.

   Hint

   Remember to check that operations modulo 5 are preserved.