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From Finite Integral Domains to Finite Fields
his article, we explore a few well-known results from abstract algebra pertaining to fields and integral domains. We ask ourselves whether every field is an integral domain, and whether every integral domain is a field.
We begin with the definition of an integral domain, discuss a few established results, and then proceed to answer these questions. This alternate proof differs in one key aspect: it does not invoke the cancellation property of integral domains stated in Proposition 2. \] Therefore every non-zero element \( a \in D \) has a multiplicative inverse in \( D. \) The remaining properties of a field are established in the same manner as in the previous section.
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