统一的本原据/h1>
有关……据/h4>
- 代数据/span>>据/span>
原始据span class="katex"> N据/span>TH.据/span>团结的根源据/英石R.O.NG>是据一种href="//www.parkandroid.com/wiki/roots-of-unity/" class="wiki_link" title="团结的根源GydF4y2Ba" target="_blank">团结的根源据/一种>谁的据一种href="//www.parkandroid.com/wiki/order-of-an-element/" class="wiki_link" title="乘法顺序GydF4y2Ba" target="_blank">乘法顺序据/一种>是据span class="katex"> N据/span>。据/span>他们是州的根源据span class="katex"> N据/span>TH.据/span>紧固多项式据/一种>,并在许多分支机构中处于中心地位据一种href="//www.parkandroid.com/wiki/number-theory/" class="wiki_link" title="数字论GydF4y2Ba" target="_blank">数字论据/一种>, 尤其据一种href="//www.parkandroid.com/wiki/algebraic-number-theory/" class="wiki_link" title="代数数字理论GydF4y2Ba" target="_blank">代数数字理论据/一种>。据/p>
定义据/h2>
让据span class="katex"> N据/span>是一个正整数。一个据英石R.O.NG>原始据span class="katex"> N据/span>TH.据/span>团结的根源据/英石R.O.NG>是一个据span class="katex"> N据/span>TH.据/span>统一的根不是据span class="katex"> K.据/span>TH.据/span>任何积极的统一的根据span class="katex"> K.据/span>据据/span>N据/span>。据/span>那是,据span class="katex"> ζ据/span>是一个原始的据span class="katex"> N据/span>TH.据/span>统一的根源,如果只有据/p>
ζ据/span>N据/span>=据/span>1据/span>那据/span>和据/span>ζ据/span>K.据/span>据/span>=据/span>1据/span>对于任何正整数据/span>K.据/span>据据/span>N据/span>。据/span>
有四个据span class="katex"> 4.据/span>TH.据/span>统一的根源据span class="katex"> ±据/span>1据/span>那据/span>±据/span>一世据/span>。据/span>其中两个,即据span class="katex"> ±据/span>一世据/span>那据/span>是原始的。另外两个不是:据span class="katex"> 1据/span>1据/span>=据/span>1据/span>和据span class="katex"> (据/span>-据/span>1据/span>)据/span>2据/span>=据/span>1据/span>。据/span>
基本属性据/h2>
原始据span class="katex"> N据/span>TH.据/span>单位的根是复数据/p>
E.据/span>2据/span>π据/span>一世据/span>K.据/span>/据/span>N据/span>:据/span>1据/span>≤.据/span>K.据/span>≤.据/span>N据/span>那据/span>肾小球囊性肾病据/span>(据/span>K.据/span>那据/span>N据/span>)据/span>=据/span>1据/span>。据/span>
有据span class="katex"> ϕ据/span>(据/span>N据/span>)据/span>原始据span class="katex"> N据/span>TH.据/span>根的统一,在哪里据span class="katex"> ϕ据/span>(据/span>N据/span>)据/span>是据一种href="//www.parkandroid.com/wiki/eulers-totient-function/" class="wiki_link" title="欧拉totient函数GydF4y2Ba" target="_blank">欧拉totient函数据/一种>。据/p>
让据span class="katex"> ζ据/span>N据/span>=据/span>E.据/span>2据/span>π据/span>一世据/span>/据/span>N据/span>。据/span>回想一下据span class="katex"> N据/span>TH.据/span>团结的根源是据span class="katex"> N据/span>明显的力量据span class="katex"> ζ据/span>N据/span>K.据/span>=据/span>E.据/span>2据/span>π据/span>一世据/span>K.据/span>/据/span>N据/span>:据/span>1据/span>≤.据/span>K.据/span>≤.据/span>N据/span>。据/span>所以它仍然表现出来据span class="katex"> ζ据/span>N据/span>K.据/span>是原始的,如果只有据span class="katex"> K.据/span>和据span class="katex"> N据/span>是副仙。据/p>
关键的事实是据span class="katex"> ζ据/span>N据/span>是一个原始的据span class="katex"> N据/span>TH.据/span>统一的根源,自首次据span class="katex"> N据/span>权力是截然不同的。标准据一种href="//www.parkandroid.com/wiki/order-of-an-element/" class="wiki_link" title="命令GydF4y2Ba" target="_blank">命令据/一种>参数显示据span class="katex"> ζ据/span>N据/span>一种据/span>=据/span>1据/span>当且仅当据span class="katex"> 一种据/span>|据/span>N据/span>那据/span>因为写作据span class="katex"> N据/span>=据/span>一种据/span>问:据/span>+据/span>R.据/span>那据/span> 0.据/span>≤.据/span>R.据/span>据据/span>N据/span>那据/span>给据span class="katex"> ζ据/span>N据/span>R.据/span>=据/span>ζ据/span>N据/span>N据/span>-据/span>一种据/span>问:据/span>=据/span>1据/span>那据/span>但除非,这是不可能的据span class="katex"> R.据/span>=据/span>0.据/span>。据/span>
如果据span class="katex"> 肾小球囊性肾病据/span>(据/span>K.据/span>那据/span>N据/span>)据/span>=据/span>D.据/span>那据/span>然后很容易检查一下据span class="katex"> (据/span>ζ据/span>N据/span>K.据/span>)据/span>N据/span>/据/span>D.据/span>=据/span>1据/span>那据/span>因此,如果据span class="katex"> D.据/span>>据/span>1据/span>然后据span class="katex"> ζ据/span>N据/span>K.据/span>不是原始的。据span class="katex"> (据/span>事实上,并不难表明据span class="katex"> ζ据/span>N据/span>K.据/span>是一个原始的据span class="katex"> (据/span>D.据/span>N据/span>)据/span>TH.据/span>团结的根。据span class="katex"> )据/span>
如果据span class="katex"> 肾小球囊性肾病据/span>(据/span>K.据/span>那据/span>N据/span>)据/span>=据/span>1据/span>那据/span>和据span class="katex"> (据/span>ζ据/span>N据/span>K.据/span>)据/span>R.据/span>=据/span>1据/span>那据/span>然后据span class="katex"> N据/span>|据/span>K.据/span>R.据/span>那据/span>但据span class="katex"> N据/span>和据span class="katex"> K.据/span>是oprime所以据span class="katex"> N据/span>|据/span>R.据/span>。据/span>这表明据span class="katex"> ζ据/span>N据/span>K.据/span>是一个原始的据span class="katex"> N据/span>TH.据/span>团结的根源,因为第一个积极的力量据span class="katex"> ζ据/span>N据/span>K.据/span>那样等于据span class="katex"> 1据/span>是据span class="katex"> (据/span>ζ据/span>N据/span>K.据/span>)据/span>N据/span>。据/span>
euler的全部函数计数正整数的数量据span class="katex"> K.据/span>≤.据/span>N据/span>它们是互质的据span class="katex"> N据/span>那据/span>这恰恰是产生原始的指数据span class="katex"> N据/span>TH.据/span>单位根,这是原始根的数目据span class="katex"> N据/span>统一的根源。据span class="katex"> □据/span>
分类的据span class="katex"> 1据/span>2据/span>TH.据/span>单位根的乘法顺序。据/p>
让据span class="katex"> ζ据/span>1据/span>2据/span>=据/span>E.据/span>2据/span>π据/span>一世据/span>/据/span>1据/span>2据/span>。据/span>然后是权力据span class="katex"> ζ据/span>1据/span>2据/span>根据指数的GCD分类据span class="katex"> 1据/span>2据/span>:据/span>如上所述,据span class="katex"> ζ据/span>1据/span>2据/span>K.据/span>是A.据span class="katex"> (据/span>G据/span>光盘据/span>(据/span>1据/span>2据/span>那据/span>K.据/span>)据/span>1据/span>2据/span>)据/span>TH.据/span>团结的根。据/p>
ζ据/span>1据/span>2据/span>那据/span>ζ据/span>1据/span>2据/span>5.据/span>那据/span>ζ据/span>1据/span>2据/span>7.据/span>那据/span>ζ据/span>1据/span>2据/span>1据/span>1据/span>是原始的据span class="katex"> 1据/span>2据/span>TH.据/span>团结的根源。据span class="katex"> (据/span>笔记据span class="katex"> ϕ据/span>(据/span>1据/span>2据/span>)据/span>=据/span>4.据/span>。据/span>)据/span>
ζ据/span>1据/span>2据/span>2据/span>那据/span>ζ据/span>1据/span>2据/span>1据/span>0.据/span>是原始的据span class="katex"> 6.据/span>TH.据/span>团结的根源。据/p>
ζ据/span>1据/span>2据/span>3.据/span>那据/span>ζ据/span>1据/span>2据/span>9.据/span>是原始的据span class="katex"> 4.据/span>TH.据/span>团结的根源。据/p>
ζ据/span>1据/span>2据/span>4.据/span>那据/span>ζ据/span>1据/span>2据/span>8.据/span>是原始的据span class="katex"> 3.据/span>rd.据/span>团结的根源。据/p>
ζ据/span>1据/span>2据/span>6.据/span>=据/span>-据/span>1据/span>是一个原始的据span class="katex"> 2据/span>n据/span>团结的根。据/p>
ζ据/span>1据/span>2据/span>1据/span>2据/span>=据/span>1据/span>是一个原始的据span class="katex"> 1据/span>英石据/span>团结的根。据/p>
请注意,订单是除数据span class="katex"> 1据/span>2据/span>。据/span>有据span class="katex"> ϕ据/span>(据/span>D.据/span>)据/span>原始根源据span class="katex"> D.据/span>那据/span>为每一个据span class="katex"> D.据/span>|据/span>1据/span>2据/span>。据/span>自据span class="katex"> D.据/span>|据/span>1据/span>2据/span>σ.据/span>ϕ据/span>(据/span>D.据/span>)据/span>=据/span>1据/span>2据/span>那据/span>这个帐户占所有人据span class="katex"> 1据/span>2据/span>TH.据/span>团结的根源。据span class="katex"> □据/span>
上面的论点表明原始的权力据span class="katex"> N据/span>TH.据/span>团结的根源枚举所有原始据span class="katex"> D.据/span>TH.据/span>统一的根,对于所有的除数据span class="katex"> D.据/span>的据span class="katex"> N据/span>。据/span>
让据span class="katex"> ζ据/span>m据/span>是一个原始的据span class="katex"> m据/span>TH.据/span>统一的根,和据br>让据span class="katex"> ζ据/span>N据/span>是一个原始的据span class="katex"> N据/span>TH.据/span>团结的根。据br>然后据span class="katex"> ζ据/span>m据/span>ζ据/span>N据/span>是一个原始的据span class="katex"> ℓ据/span>TH.据/span>一些正整数的统一根据span class="katex"> ℓ据/span>。据/span>
我们能说什么据span class="katex"> ℓ据/span>一般来说?据/p>
澄清:据/英石R.O.NG>在答案选择中,据span class="katex">
G据/span>光盘据/span>(据/span>⋅据/span>)据/span>和据span class="katex">
LCM.据/span>(据/span>⋅据/span>)据/span>分别表示最大的常见除法函数和最低的常见多功能。据/p>
金额和产品据/h2>
原始的产物据span class="katex"> N据/span>TH.据/span>统一的根源是据span class="katex"> 1据/span>除非据span class="katex"> N据/span>=据/span>2据/span>。据/span>这是因为原始的一组据span class="katex"> N据/span>TH.据/span>统一的根,据span class="katex"> N据/span>≥据/span>3.据/span>那据/span>可以分成对吗据span class="katex"> {据/span>ζ据/span>K.据/span>那据/span>ζ据/span>N据/span>-据/span>K.据/span>}据/span>那据/span>哪个乘以给予据span class="katex"> 1据/span>。据/span> (据/span>为了据span class="katex"> N据/span>=据/span>2据/span>这失败了,因为据span class="katex"> ζ据/span>1据/span>和据span class="katex"> ζ据/span>2据/span>-据/span>1据/span>重合。据span class="katex"> )据/span>
原始的总和据span class="katex">
N据/span>TH.据/span>统一的根源是据span class="katex">
μ据/span>(据/span>N据/span>)据/span>那据/span>在哪里据span class="katex">
μ据/span>是个据一种href="//www.parkandroid.com/wiki/mobius-function/" class="wiki_link" title="Möbius函数GydF4y2Ba" target="_blank">Möbius函数据/一种>;看到Wiki的证据。据/p>
所有原始的和是什么据span class="katex">
2据/span>0.据/span>1据/span>5.据/span>TH.据/span>统一的根,据span class="katex">
W.据/span>,意思是2015年是最小的正整数据span class="katex">
N据/span>这样据span class="katex">
W.据/span>N据/span>=据/span>1据/span>还是据/p>
额外的信用问题:原始的总和是多少据span class="katex">
2据/span>0.据/span>0.据/span>9.据/span>TH.据/span>统一的根源?据/p>
事实上,有优雅的公式据span class="katex">
K.据/span>TH.据/span>原始的力量据span class="katex">
N据/span>TH.据/span>团结的根源:据/p>
和据span class="katex">
K.据/span>TH.据/span>原始的力量据span class="katex">
N据/span>TH.据/span>统一的根源是据/p>
μ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>N据/span>)据/span>那据/span> 在哪里据span class="katex">
R.据/span>=据/span>G据/span>光盘据/span>(据/span>N据/span>那据/span>K.据/span>)据/span>N据/span>。据/span> 定理的极端例子是什么时候据span class="katex">
肾小球囊性肾病据/span>(据/span>N据/span>那据/span>K.据/span>)据/span>=据/span>1据/span>什么时候据span class="katex">
肾小球囊性肾病据/span>(据/span>N据/span>那据/span>K.据/span>)据/span>=据/span>N据/span>。据/span> 当据span class="katex">
肾小球囊性肾病据/span>(据/span>N据/span>那据/span>K.据/span>)据/span>=据/span>1据/span>那据/span>服用据span class="katex">
K.据/span>TH.据/span>幂对原语进行置换据span class="katex">
N据/span>TH.据/span>单位根,所以和仍然是据span class="katex">
μ据/span>(据/span>N据/span>)据/span>。据/span>的确,据span class="katex">
R.据/span>=据/span>N据/span>那据/span>所以据span class="katex">
μ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>N据/span>)据/span>=据/span>μ据/span>(据/span>N据/span>)据/span>。据/span> 当据span class="katex">
肾小球囊性肾病据/span>(据/span>N据/span>那据/span>K.据/span>)据/span>=据/span>N据/span>那据/span>权力都是据span class="katex">
1据/span>那据/span>所以总和是据span class="katex">
ϕ据/span>(据/span>N据/span>)据/span>。据/span>在这种情况下据span class="katex">
R.据/span>=据/span>1据/span>那据/span>所以据span class="katex">
μ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>N据/span>)据/span>=据/span>ϕ据/span>(据/span>N据/span>)据/span>如预期的。据/p>
这是一个大纲:总和据span class="katex">
σ.据/span>ζ据/span>K.据/span>是原始的和吗据span class="katex">
R.据/span>TH.据/span>统一的根源,它遍布所有人。但有重复:总和有据span class="katex">
ϕ据/span>(据/span>N据/span>)据/span>条款和有据span class="katex">
ϕ据/span>(据/span>R.据/span>)据/span>原始据span class="katex">
R.据/span>TH.据/span>统一的根源,所以它们都是计数的据span class="katex">
ϕ据/span>(据/span>R.据/span>)据/span>ϕ据/span>(据/span>N据/span>)据/span>时代。原始的总和据span class="katex">
R.据/span>TH.据/span>统一的根源是据span class="katex">
μ据/span>(据/span>R.据/span>)据/span>那据/span>结果就是这样。据span class="katex">
□据/span>
组结构据/h2>
这据span class="katex"> 1据/span>2据/span>TH.据/span>统一的根是同构的据span class="katex"> Z.据/span>/据/span>1据/span>2据/span>。据/span>原始据span class="katex"> 1据/span>2据/span>TH.据/span>统一的根,据span class="katex"> ζ据/span>1据/span>2据/span>那据/span>ζ据/span>1据/span>2据/span>5.据/span>那据/span>ζ据/span>1据/span>2据/span>7.据/span>那据/span>和据span class="katex"> ζ据/span>1据/span>2据/span>1据/span>1据/span>那据/span>对应于元素据span class="katex"> 1据/span>那据/span>5.据/span>那据/span>7.据/span>那据/span>1据/span>1据/span>在据span class="katex"> Z.据/span>/据/span>1据/span>2据/span>那据/span>分别。这些是生成的元素据span class="katex"> Z.据/span>/据/span>1据/span>2据/span>加剧。例如。倍数据span class="katex"> 5.据/span>是据/p>
5.据/span>那据/span>1据/span>0.据/span>那据/span>3.据/span>那据/span>8.据/span>那据/span>1据/span>那据/span>6.据/span>那据/span>1据/span>1据/span>那据/span>4.据/span>那据/span>9.据/span>那据/span>2据/span>那据/span>7.据/span>那据/span>0.据/span>那据/span>5.据/span>那据/span>1据/span>0.据/span>那据/span>......据/span>。据/span>
每一个元素据span class="katex"> Z.据/span>/据/span>1据/span>2据/span>在那个列表。据/p>
紧固多项式据/h2>
多项式据span class="katex-display"> ζ据/span>一个原始的据/span>N据/span>单位的Th根据/span>π据/span>(据/span>X据/span>-据/span>ζ据/span>)据/span>是一个多项式据span class="katex"> X据/span>被称为据span class="katex"> N据/span>TH.据一种href="//www.parkandroid.com/wiki/cyclotomic-polynomials/" class="wiki_link" title="紧固多项式GydF4y2Ba" target="_blank">紧固多项式据/一种>。它非常兴趣据一种href="//www.parkandroid.com/wiki/algebraic-number-theory/" class="wiki_link" title="代数数字理论GydF4y2Ba" target="_blank">代数数字理论据/一种>。有关更多详细信息和属性,请参阅Wiki On据一种href="//www.parkandroid.com/wiki/cyclotomic-polynomials/" class="wiki_link" title="紧固多项式GydF4y2Ba" target="_blank">紧固多项式据/一种>。据/p>