24.3.2.1. Proof by Contradiction

Try these first

1. 1.

   What does the characteristic of a field represent?

   Hint

   Think about addition and returning to zero.

2. 2.

   Define what a prime number is.

   Hint

   Consider how many factors it has.

3. 3.

   What denotes the characteristic of a field?

   • The sum of all elements
   • The additive order of multiplicative identity
   • The highest number in the field

   Hint

   Think of how elements interact in addition.

4. 4.

   True or False: The characteristic of any finite field must be a prime number.

   • True
   • False

   Hint

   Recall the definitions and the implications of composite numbers.

5. 5.

   Find the characteristic of the finite field containing the elements {0, 1, 2} with operations defined by addition and multiplication modulo 3.

   Hint

   Focus on how the operations cycle through the elements.

6. 6.

   If we assume a field has a characteristic of 6, demonstrate how you would derive a contradiction.

   Hint

   Break the assumption into cases for each factor of 6.