| 21. | Therefore there are subfields nested inside the cyclotomic field, each of degree 2 over the one before.
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| 22. | They have a simple algebraic description but no obvious cyclotomic properties which special purpose algorithms can exploit ."
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| 23. | This may be seen by using the Kronecker Weber theorem to embed the abelian field into a cyclotomic field.
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| 24. | Already Gauss had shown that, in fact, every quadratic field is contained in a larger cyclotomic field.
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| 25. | The cyclotomic units form a subgroup of finite class number of the maximal real subfield of the cyclotomic field.
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| 26. | The cyclotomic units form a subgroup of finite class number of the maximal real subfield of the cyclotomic field.
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| 27. | Such transform can be carried out efficiently with proper fast algorithms, for example, cyclotomic fast Fourier transform.
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| 28. | Other choices of n are possible but the corresponding cyclotomic polynomials are more complicated or their security not as well studied.
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| 29. | These can be at least partially factored using cyclotomic polynomials, although I do not know what algorithm the calculator uses.
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| 30. | In particular cases, for example when is a cyclotomic extension of, there are particular and more refined class number formulas.
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