oxeimon
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Is there an easy way to define a field by only using one operation?
It's easy to see that a cyclic abelian group of prime order uniquely defines a field if you consider multiplication as just repeated addition. Ie, consider such a group G with generator g, identity e with order p. Then, we can say G = \{g,2g,3g,\ldots,(p-1)g,pg = e\}.
Just define multiplication as x\cdot y = \sum_{i=1}^jx where j is the order of y with respect to the generator _g. This is clearly commutative, and inverses exist because p is prime.
Is there a similarly easy way to extend this to polynomial fields as well?
Also, why are my singleton letters so high up?
It's easy to see that a cyclic abelian group of prime order uniquely defines a field if you consider multiplication as just repeated addition. Ie, consider such a group G with generator g, identity e with order p. Then, we can say G = \{g,2g,3g,\ldots,(p-1)g,pg = e\}.
Just define multiplication as x\cdot y = \sum_{i=1}^jx where j is the order of y with respect to the generator _g. This is clearly commutative, and inverses exist because p is prime.
Is there a similarly easy way to extend this to polynomial fields as well?
Also, why are my singleton letters so high up?
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