Suppose that ≺D,+,⋅≻ is a well-ordered integral domain. Then the least positive element in D is the unity e∈D, since if a∈Dp such that e>a, then e−a∈Dp⟹a(e−a)∈Dp⟹a−a2∈Dp. I.

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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Suppose that ≺D,+,⋅≻ is a well-ordered integral domain. Then the least positive element in D is the unity e∈D, since if a∈Dp such that e>a, then e−a∈Dp⟹a(e−a)∈Dp⟹a−a2∈Dp. I.e., e>a>a2, contradiction.

 
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Step 1: Step 1

Suppose that ≺D,+,⋅≻ is a well-ordered integral domain and let e be the least positive element in D.

  If there exists an element a∈D+ such that e>a, then e−a∈D+ and a(e−a)∈D+

But a(ea)=aa2 ,

so we have e>a>a2

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