Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be an integer, let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> be a prime number, let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1<g<p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>g</mi> <mo><</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> be a primitive root modulo p, and let F denote either the quadratic field <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{p})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>p</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the case n is even and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p\equiv 1 \pmod {4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{-p})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mo>-</mo> <mi>p</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the case n is odd and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p\equiv 3 \pmod {4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>3</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the class number of the field F is expressed in terms of the alternating sum of the digits of the repetend in the g-adic expansion of 1/p in Girstmair (Am Math Mon 101(10):997–1001, 1994; Acta Arith 67(4):381–386, 1994), and Murty and Thangadurai (Proc Amer Math Soc 139(4):1277–1289, 2011). In this paper, the orders of even algebraic K-groups <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$K_{2n-2}(\mathcal {O}_F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">O</mi> <mi>F</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for the ring of integers <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathcal {O}_F$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation> of F, are expressed in terms of the digits of the repetend of the g-adic expansion of 1/p for any integer <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Algebraic K-groups of $\mathbb {Q}(\sqrt{\pm p})$ and the digits of 1/p

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2026年6月25日

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Reza Taleb

摘要

Let $n\ge 1$ $n \geq 1$ be an integer, let $p\ge 5$ $p \geq 5$ be a prime number, let $1<g<p$ $1 < g < p$ be a primitive root modulo p, and let F denote either the quadratic field $\mathbb {Q}(\sqrt{p})$ $Q (\sqrt{p})$ in the case n is even and $p\equiv 1 \pmod {4}$ $p \equiv 1 (mod 4)$ , or $\mathbb {Q}(\sqrt{-p})$ $Q (\sqrt{- p})$ in the case n is odd and $p\equiv 3 \pmod {4}$ $p \equiv 3 (mod 4)$ . For $n=1$ $n = 1$ , the class number of the field F is expressed in terms of the alternating sum of the digits of the repetend in the g-adic expansion of 1/p in Girstmair (Am Math Mon 101(10):997–1001, 1994; Acta Arith 67(4):381–386, 1994), and Murty and Thangadurai (Proc Amer Math Soc 139(4):1277–1289, 2011). In this paper, the orders of even algebraic K-groups $K_{2n-2}(\mathcal {O}_F)$ $K_{2 n - 2} (O_{F})$ , for the ring of integers $\mathcal {O}_F$ $O_{F}$ of F, are expressed in terms of the digits of the repetend of the g-adic expansion of 1/p for any integer $n\ge 2$ $n \geq 2$ .

Algebraic K-groups of \(\mathbb {Q}(\sqrt{\pm p})\) and the digits of 1/p

摘要