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The translate and line properties for 2-primitive elements in quadratic extensions
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2020-06-16 , DOI: 10.1142/s1793042120501055 Stephen D. Cohen 1 , Giorgos Kapetanakis 2
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2020-06-16 , DOI: 10.1142/s1793042120501055 Stephen D. Cohen 1 , Giorgos Kapetanakis 2
Affiliation
Let [Formula: see text] be integers and [Formula: see text] be any prime power [Formula: see text] such that [Formula: see text]. We say that the extension [Formula: see text] possesses the line property for [Formula: see text]-primitive elements if, for every [Formula: see text], such that [Formula: see text], there exists some [Formula: see text], such that [Formula: see text] has multiplicative order [Formula: see text]. Likewise, if, in the above definition, [Formula: see text] is restricted to the value [Formula: see text], we say that [Formula: see text] possesses the translate property. In this paper, we take [Formula: see text] (so that necessarily [Formula: see text] is odd) and prove that [Formula: see text] possesses the translate property for 2-primitive elements unless [Formula: see text]. With some additional theoretical and computational effort, we show also that [Formula: see text] possesses the line property for 2-primitive elements unless [Formula: see text].
中文翻译:
二次扩展中 2 基元元素的 translate 和 line 属性
令 [Formula: see text] 为整数, [Formula: see text] 为任意素数 [Formula: see text] 使得 [Formula: see text]。我们说扩展 [Formula: see text] 拥有 [Formula: see text]-原始元素的 line 属性,如果对于每个 [Formula: see text],使得 [Formula: see text] 存在一些 [Formula: see text] : 见正文],使得 [公式:见正文] 具有乘法顺序 [公式:见正文]。同样,如果在上面的定义中,[Formula: see text] 被限制为 [Formula: see text] 的值,我们说 [Formula: see text] 拥有 translate 属性。在本文中,我们取 [Formula: see text](因此 [Formula: see text] 必然是奇数)并证明 [Formula: see text] 具有 2-primitive 元素的 translate 属性,除非 [Formula: see text] .
更新日期:2020-06-16
中文翻译:
二次扩展中 2 基元元素的 translate 和 line 属性
令 [Formula: see text] 为整数, [Formula: see text] 为任意素数 [Formula: see text] 使得 [Formula: see text]。我们说扩展 [Formula: see text] 拥有 [Formula: see text]-原始元素的 line 属性,如果对于每个 [Formula: see text],使得 [Formula: see text] 存在一些 [Formula: see text] : 见正文],使得 [公式:见正文] 具有乘法顺序 [公式:见正文]。同样,如果在上面的定义中,[Formula: see text] 被限制为 [Formula: see text] 的值,我们说 [Formula: see text] 拥有 translate 属性。在本文中,我们取 [Formula: see text](因此 [Formula: see text] 必然是奇数)并证明 [Formula: see text] 具有 2-primitive 元素的 translate 属性,除非 [Formula: see text] .