The main aim of this paper is to compute the unit group of some fields of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {L}^+=\mathbb {Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">L</mi> </mrow> <mo>+</mo> </msup> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <msqrt> <mn>2</mn> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">pq</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">ps</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {L}=\mathbb {Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell }),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">L</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mn>2</mn> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">pq</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">ps</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mo>-</mo> <mi>ℓ</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where q, p, s are three different prime integers satisfying certain conditions and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is a positive odd square-free integer relatively prime to q, p and s.

Fundamental system of units of some multiquadratic fields of degree 8 and degree 16

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2025年9月22日

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Moha Ben Taleb El Hamam

摘要

The main aim of this paper is to compute the unit group of some fields of the form $\mathbb {L}^+=\mathbb {Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ $L^{+} = Q (\sqrt{2}, \sqrt{pq}, \sqrt{ps})$ and $\mathbb {L}=\mathbb {Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{-\ell }),$ $L = Q (\sqrt{2}, \sqrt{pq}, \sqrt{ps}, \sqrt{- ℓ}),$ where q, p, s are three different prime integers satisfying certain conditions and $\ell \ge 1$ $ℓ \geq 1$ is a positive odd square-free integer relatively prime to q, p and s.