WOLFRAM

Wolfram 语言与系统参考资料中心

AlgebraicUnitQ[a]

如果 a 为代数单位则返回 True，否则返回 False.

更多信息
Details and Options Details and Options
范例  
基本范例  
范围  
应用  
基本应用  
数论  
属性和关系  
可能存在的问题  
参见
技术笔记
相关指南
历史
按以下格式引用：

AlgebraicUnitQ

AlgebraicUnitQ[a]

如果 a 为代数单位则返回 True，否则返回 False.

更多信息

  • AlgebraicUnitQ 通常用于测试一个数是否为代数单位.
  • 代数单位 a 是 a 和 1/a 均为代数整数的数值.
  • 除非 a 明显是代数单位，否则 AlgebraicUnitQ[a] 返回 False.

范例

打开所有单元 关闭所有单元

基本范例  (2)

检验一个数是否为代数单位：

Wolfram Language code: AlgebraicUnitQ[I]

数 不是代数单位：

Wolfram Language code: AlgebraicUnitQ[Sqrt[2]]

范围  (4)

AlgebraicUnitQ 适用于整数：

Wolfram Language code: AlgebraicUnitQ[1]

实数：

Wolfram Language code: AlgebraicUnitQ[2.4]

复数：

Wolfram Language code: AlgebraicUnitQ[Exp[I 2Pi / 5]]

超越数：

Wolfram Language code: AlgebraicUnitQ[Pi]

根表达式：

Wolfram Language code: AlgebraicUnitQ[5 ^ (2 / 3) + 7 ^ (1 / 5) + 10]

Root 对象：

Wolfram Language code: AlgebraicUnitQ[Root[1 + #1 + #1^2 + #1^3 + #1^4 + #1^5 + #1^6&, 1]]

AlgebraicNumber 对象：

Wolfram Language code: AlgebraicUnitQ[AlgebraicNumber[Root[#1 ^ 3 - 4 #1 + 17&, 1], {1, 2, 3}]]

AlgebraicUnitQ 线性作用于列表：

Wolfram Language code: AlgebraicUnitQ[{I, Sqrt[2], 1 / Sqrt[5]}]

应用  (6)

基本应用  (1)

生成随机代数单位：

Wolfram Language code: randomAlgebraicUnit[n_, m_ : 1] := Table[Root[Dot[Table[1, n + 1], # ^ Range[0, n]], RandomInteger[{1, n}]], {m}];
Wolfram Language code: randomAlgebraicUnit[8, 10]
Wolfram Language code: AlgebraicUnitQ[%]

绘制代数单位：

Wolfram Language code: ComplexListPlot[randomAlgebraicUnit[100, 50]]

数论  (5)

求是代数单位的整数：

Wolfram Language code: x /. Solve[x * y == 1, {x, y}, Integers]
Wolfram Language code: {AlgebraicUnitQ[1], AlgebraicUnitQ[-1]}

所有的单位根都是代数单位：

Wolfram Language code: Exp[(2Pi Range[0, 6]I) / 6]
Wolfram Language code: AlgebraicUnitQ[%]

代数单位的范数为 或 ：

Wolfram Language code: AlgebraicUnitQ[GoldenRatio]
Wolfram Language code: AlgebraicNumberNorm[GoldenRatio]

中范数 的表示：

Wolfram Language code: {a} = NumberFieldNormRepresentatives[Sqrt[5], 5]
Wolfram Language code: AlgebraicNumberNorm[a]

可以用一个单位乘以 a 来表示：

Wolfram Language code: AlgebraicUnitQ[-2 - Sqrt[5]]
Wolfram Language code: RootReduce[Sqrt[5](-2 - Sqrt[5]) == a]

使用单位根求 Cyclotomic 多项式：

Wolfram Language code: roots = Exp[2 Pi I / Range[3]]
Wolfram Language code: MinimalPolynomial[roots, x]
Wolfram Language code: Cyclotomic[Range[3], x]

属性和关系  (7)

一个代数单位及其倒数为代数整数：

Wolfram Language code: a = 1 - 6 6 ^ (1 / 3) + 3 36 ^ (1 / 3);
Wolfram Language code: AlgebraicIntegerQ [{a, 1 / a}]

该倒数是一个代数单位：

Wolfram Language code: AlgebraicUnitQ [{a, 1 / a}]

既是代数整数又是代数单位的整数仅有 和 ：

Wolfram Language code: {AlgebraicIntegerQ[1], AlgebraicUnitQ[1]}
Wolfram Language code: {AlgebraicIntegerQ[-1], AlgebraicUnitQ[-1]}

一个代数单位自乘一个幂也是一个代数单位：

Wolfram Language code: Table[AlgebraicUnitQ[(1 + Sqrt[2]) ^ n], {n, 5}]

Algebraics 表示所有代数数域，包括代数单位：

Wolfram Language code: Element[Exp[Pi I / 2], Algebraics]

代数单位具有范数 或 ：

Wolfram Language code: AlgebraicUnitQ[{GoldenRatio, Exp[Pi I / 2]}]
Wolfram Language code: AlgebraicNumberNorm[{GoldenRatio, Exp[Pi I / 2]}]

使用 MinimalPolynomial 求代数单位的最小多项式：

Wolfram Language code: MinimalPolynomial[4 / 5, x]

现在求其倒数的最小多项式；

Wolfram Language code: MinimalPolynomial[5 / 4, x]

数域的基本单位是代数单位：

Wolfram Language code: NumberFieldFundamentalUnits[Root[-1 + 2#1 ^ 4&, 1]]
Wolfram Language code: AlgebraicUnitQ[%]

可能存在的问题  (1)

在某些情况下，一个数是否为代数单位不可知：

Wolfram Language code: AlgebraicUnitQ[E + Pi]
Wolfram Research (2007)，AlgebraicUnitQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html.

文本

Wolfram Research (2007)，AlgebraicUnitQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html.

CMS

Wolfram 语言. 2007. "AlgebraicUnitQ." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html.

APA

Wolfram 语言. (2007). AlgebraicUnitQ. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html 年

BibTeX

@misc{reference.wolfram_2026_algebraicunitq, author="Wolfram Research", title="{AlgebraicUnitQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html}", note=[Accessed: 16-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_algebraicunitq, organization={Wolfram Research}, title={AlgebraicUnitQ}, year={2007}, url={https://reference.wolfram.com/language/ref/AlgebraicUnitQ.html}, note=[Accessed: 16-September-2026]}