Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_{q^m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>m</mi> </msup> </msub> </math></EquationSource> </InlineEquation> denote the degree-m extension of the finite field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>, where q is a prime power and m is a positive integer. Suppose <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$r_1, r_2 \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k_1, k_2 \in \mathbb {N} \cup \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>∪</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$r_1, r_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> divide <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q^m - 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$F = \frac{F_1}{F_2} \in \mathbb {F}_{q^m}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>=</mo> <mfrac> <msub> <mi>F</mi> <mn>1</mn> </msub> <msub> <mi>F</mi> <mn>2</mn> </msub> </mfrac> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>m</mi> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be a rational function subject to minor conditions. In this article, for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a, b \in \mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, we establish a sufficient condition on the pair (q, m) that guarantees the existence of an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$r_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-primitive <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$k_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-normal element <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\alpha \in \mathbb {F}_{q^m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>m</mi> </msup> </msub> </mrow> </math></EquationSource> </InlineEquation> such that trace of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> is a, trace of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\alpha ^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>α</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathbb {F}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> is b and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$F(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$r_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-primitive <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$k_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-normal element in <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\mathbb {F}_{q^m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>m</mi> </msup> </msub> </math></EquationSource> </InlineEquation>.

Rational image of r-primitive k-normal elements with prescribed traces

期刊论文
发表日期：

2026年5月2日

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A. Choudhary,
J. Sharma,
S. K. Tiwari,
K. Chatterjee

摘要

Let $\mathbb {F}_{q^m}$ $F_{q^{m}}$ denote the degree-m extension of the finite field $\mathbb {F}_q$ $F_{q}$ , where q is a prime power and m is a positive integer. Suppose $r_1, r_2 \in \mathbb {N}$ $r_{1}, r_{2} \in N$ and $k_1, k_2 \in \mathbb {N} \cup \{0\}$ $k_{1}, k_{2} \in N \cup {0}$ are such that $r_1, r_2$ $r_{1}, r_{2}$ divide $q^m - 1$ $q^{m} - 1$ . Let $F = \frac{F_1}{F_2} \in \mathbb {F}_{q^m}(x)$ $F = \frac{F_{1}}{F_{2}} \in F_{q^{m}} (x)$ be a rational function subject to minor conditions. In this article, for $a, b \in \mathbb {F}_q$ $a, b \in F_{q}$ , we establish a sufficient condition on the pair (q, m) that guarantees the existence of an $r_1$ $r_{1}$ -primitive $k_1$ $k_{1}$ -normal element $\alpha \in \mathbb {F}_{q^m}$ $α \in F_{q^{m}}$ such that trace of $\alpha $ $α$ over $\mathbb {F}_{q}$ $F_{q}$ is a, trace of $\alpha ^{-1}$ $α^{- 1}$ over $\mathbb {F}_{q}$ $F_{q}$ is b and $F(\alpha )$ $F (α)$ is an $r_2$ $r_{2}$ -primitive $k_2$ $k_{2}$ -normal element in $\mathbb {F}_{q^m}$ $F_{q^{m}}$ .