<p>We provide an infinite collection of exceptional quartic number fields by specifying exceptional units <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> of degree 4 by means of their minimal polynomials. Along with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, we show that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha ^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>α</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> is also an exceptional unit. Our collection of fields is rich enough to have any real quadratic field as a subfield of one of them. On the other hand, another infinite collection of quartic exceptional fields with no quadratic subfields is also provided. In both the collections, the fields have unit rank 3, they are non-Galois extensions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">Q</mi> </math></EquationSource> </InlineEquation>, and their normal closures have Galois groups <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$D_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$S_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>, respectively. We further show that one can find a vast collection of higher degree exceptional number fields as an easy consequence of Perron’s irreducibility theorem.</p>

Exceptional number fields: quartics and higher degree

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发表日期：

2026年5月25日

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C Aruna,
P Vanchinathan

摘要

We provide an infinite collection of exceptional quartic number fields by specifying exceptional units $\alpha $ $α$ of degree 4 by means of their minimal polynomials. Along with $\alpha $ $α$ , we show that $\alpha ^2$ $α^{2}$ is also an exceptional unit. Our collection of fields is rich enough to have any real quadratic field as a subfield of one of them. On the other hand, another infinite collection of quartic exceptional fields with no quadratic subfields is also provided. In both the collections, the fields have unit rank 3, they are non-Galois extensions of $\textbf{Q}$ $Q$ , and their normal closures have Galois groups $D_4$ $D_{4}$ and $S_4$ $S_{4}$ , respectively. We further show that one can find a vast collection of higher degree exceptional number fields as an easy consequence of Perron’s irreducibility theorem.