Not only is the symbol daunting, but the words are too

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SUMMARY

The discussion centers on verifying that the set \mathbb{F}_2 = {0, 1} is a field by checking its properties. Participants emphasize the importance of demonstrating the axioms A1 to A4 and M1 to M4, which define field operations, including closure, identity elements, and inverses. The term "curious field" refers to \mathbb{F}_2 being the smallest field, where operations like 1 + 1 = 0 illustrate unique characteristics of binary arithmetic. The conversation highlights the necessity of understanding these properties to confirm that \mathbb{F}_2 meets the criteria for a field.

PREREQUISITES
  • Understanding of field theory and its axioms (A1 to A4, M1 to M4).
  • Familiarity with binary operations and their properties.
  • Knowledge of additive and multiplicative identities in algebra.
  • Basic concepts of closure and inverses in mathematical structures.
NEXT STEPS
  • Study the properties of finite fields, particularly \mathbb{F}_p for prime p.
  • Learn how to construct addition and multiplication tables for small fields.
  • Explore the implications of closure under addition and multiplication in field theory.
  • Investigate the concept of additive and multiplicative inverses in various fields.
USEFUL FOR

Mathematics students, educators, and anyone interested in abstract algebra, particularly those studying field theory and its applications in various mathematical contexts.

  • #61
This is probably another stupid question.

Since M3 states the same condition, it would be wrong/unnecessary to even do 0*1 = 0 because it contradicts M4 \ {0} right?
 
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  • #62
M3 does not state the same thing as M4...
 
  • #63
micromass said:
M3 does not state the same thing as M4...

But they have a similar condition...
 
  • #64
Oh wait you know what, never mind. I was looking at 2.49 and I thought 0^{-1} = 0 and I had to test 0*1 = 0.

So yes I do need to test 0*1 = 0 for x =0
 

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