19.2.8. Field Axioms

Try these first

1. 1.

   What defines a field in contrast to a ring?

   Hint

   Think about whether every element has an inverse.

2. 2.

   Name a common example of a field.

   Hint

   Consider numbers that can be expressed as a fraction.

3. 3.

   Which of the following is not a requirement for a field?

   • Every element must have an additive inverse
   • Every non-zero element must have a multiplicative inverse
   • All elements must be even

   Hint

   Focus on the specific definitions of field properties.

4. 4.

   True or False: In a field, if ab = 0, then a = 0 or b = 0.

   • True
   • False

   Hint

   Consider how products work in different structures.

5. 5.

   Using the properties of fields, demonstrate why the set of complex numbers forms a field.

   Hint

   Break down numbers into real and imaginary parts to confirm inverses.

6. 6.

   Let F be a field and let P(x) be a polynomial in F[x]. If P(x) has degree n, show that P(x) can be expressed in terms of its coefficients in F.

   Hint

   Recall the polynomial structure and its relation to field elements.