Field/Basic properties/Fact/Proof2

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  1. We have . Subtracting (meaning addition with the negative of ) on both sides gives the claim.
  2. See exercise.
  3. See exercise.
  4. See exercise.
  5. We prove this by contradiction, so we assume that and are both not . Then there exist inverse elements and and hence . On the other hand, we have by the premise and so the annulation rule gives

    hence , which contradicts the field properties.

  6. This follows with a double induction, see exercise.