有限田地据/h1>
已经有一个帐户?据一种href="//www.parkandroid.com/account/login/?next=/wiki/finite-fields/" class="ax-click" data-ax-id="clicked_signup_modal_login" data-ax-type="link">这里登录。据/一种>据/P.>据/D.一世v>
理论据strong>有限田地据/strong>是一个关键部分据一种href="//www.parkandroid.com/wiki/number-theory/" class="wiki_link" title="数字论GydF4y2Ba" target="_blank">数字论据/一种>那据一种href="//www.parkandroid.com/wiki/abstract-algebra/" class="wiki_link" title="抽象代数GydF4y2Ba" target="_blank">抽象代数据/一种>,算术据一种href="//www.parkandroid.com/wiki/algebraic-geometry/" class="wiki_link" title="代数几何GydF4y2Ba" target="_blank">代数几何据/一种>, 和据一种href="//www.parkandroid.com/wiki/cryptography/" class="wiki_link" title="加密GydF4y2Ba" target="_blank">加密据/一种>等等。很多关于这个问题的问题据一种href="//www.parkandroid.com/wiki/integers/" class="wiki_link" title="整数GydF4y2Ba" target="_blank">整数据/一种>或者据一种href="//www.parkandroid.com/wiki/rational-numbers/" class="wiki_link" title="有理数GydF4y2Ba" target="_blank">有理数据/一种>可以转化为有限领域中算术的问题,这往往更具易行。因为有限的领域很适合计算机计算,它们在许多现代使用据一种href="//www.parkandroid.com/wiki/cryptography/" class="wiki_link" title="加密应用程序GydF4y2Ba" target="_blank">加密应用程序据/一种>。据/P.>据/D.一世v>
内容据/h4>
定义和例子据/h2>
一种据strong>场地据/strong>是A.据一种href="//www.parkandroid.com/wiki/ring-theory/" class="wiki_link" title="换向戒指GydF4y2Ba" target="_blank">换向戒指据/一种>其中每个非零元素都有乘法反向。据B.r>也就是说,一个字段是一组据span class="katex"> F据/span>有两项操作,据span class="katex"> +据/span>和据span class="katex"> ⋅据/span>,这样据/P.>据P.>(1)据span class="katex"> F据/span>是一个据一种href="//www.parkandroid.com/wiki/group-theory-introduction/" class="wiki_link" title="阿贝利亚集团GydF4y2Ba" target="_blank">阿贝利亚集团据/一种>补充;据B.r>(2)据span class="katex"> F据/span>*据/span>=据/span>F据/span>-据/span>{据/span>0.据/span>}据/span>是一群乘法,在哪里据span class="katex"> 0.据/span>是添加剂的身份据span class="katex"> F据/span>;据B.r>(3)据span class="katex"> 一种据/span>⋅据/span>(据/span>B.据/span>+据/span>C据/span>)据/span>=据/span>一种据/span>⋅据/span>B.据/span>+据/span>一种据/span>⋅据/span>C据/span>对所有人据span class="katex"> 一种据/span>那据/span>B.据/span>那据/span>C据/span>∈据/span>F据/span>。据/P.>据!-- end-definition -->
有理数据span class="katex"> 问:据/span>,实数据span class="katex"> R.据/span>,以及复数据span class="katex"> C据/span>是所有领域。据/P.>据P.>如果据span class="katex"> α.据/span>是一个据一种href="//www.parkandroid.com/wiki/algebraic-number-theory/" class="wiki_link" title="代数整数GydF4y2Ba" target="_blank">代数整数据/一种>, 然后据span class="katex"> 问:据/span>(据/span>α.据/span>)据/span>也是一个领域。据/P.>据P.>这据一种href="//www.parkandroid.com/wiki/ring-theory/" class="wiki_link" title="商戒指GydF4y2Ba" target="_blank">商戒指据/一种>据span class="katex"> Z.据/span>/据/span>(据/span>P.据/span>)据/span>是一个领域据span class="katex"> P.据/span>元素。(这有时是写的据span class="katex"> Z.据/span>P.据/span>但是,大多数教科书为不同的环涉及的戒指储备了此表示法据span class="katex"> P.据/span>-Adic Integers。)据/P.>据!-- end-example -->
同构和特征据/h2>
最后一个例子是有限的字段据span class="katex">
P.据/span>元素。从现在开始,它将被写据span class="katex">
F据/span>P.据/span>并称为“与之的领域据span class="katex">
P.据/span>元素。“这需要一些理由:为什么只有一个字段据span class="katex">
P.据/span>元素?为了理解这一点,有必要定义两个字段相同的概念。据/P.>据B.lockquote class="definition">
一种据strong>同性恋据/strong>两个戒指之间据span class="katex">
F据/span>和据span class="katex">
G据/span>是一个功能据span class="katex">
F据/span>:据/span>F据/span>→据/span>G据/span>这样据/P.>据P.>(1)据span class="katex">
F据/span>(据/span>一种据/span>+据/span>F据/span>B.据/span>)据/span>=据/span>F据/span>(据/span>一种据/span>)据/span>+据/span>G据/span>F据/span>(据/span>B.据/span>)据/span> (2)据span class="katex">
F据/span>(据/span>一种据/span>×据/span>F据/span>B.据/span>)据/span>=据/span>F据/span>(据/span>一种据/span>)据/span>×据/span>G据/span>F据/span>(据/span>B.据/span>)据/span> (3)据span class="katex">
F据/span>(据/span>1据/span>F据/span>)据/span>=据/span>1据/span>G据/span>。据/span> 一个据strong>同构据/strong>是一种同性恋也是一个据一种href="//www.parkandroid.com/wiki/bijective-functions/" class="wiki_link" title="biGydF4y2Ba" target="_blank">bi据/一种>。如果两个环之间存在同构,则据说戒指据strong>同构据/strong>。据!-- end-definition --> 在一个领域据span class="katex">
F据/span>, 让据span class="katex">
P.据/span>是最小的正整数,这样据span class="katex">
P.据/span>⋅据/span>1据/span>=据/span>P.据/span>时代据/span>
1据/span>+据/span>1据/span>+据/span>⋯据/span>+据/span>1据/span>=据/span>0.据/span>在据span class="katex">
F据/span>。如果据span class="katex">
P.据/span>不存在,说据span class="katex">
F据/span>已据strong>特征据span class="katex">
0.据/span>, 而如果据span class="katex">
P.据/span>存在,说据span class="katex">
F据/span>已据strong>特征据span class="katex">
P.据/span>。据/P.>据!-- end-definition -->
特征有助于分类有限字段:据/P.>据B.lockquote>
如果据span class="katex">
F据/span>有特色据span class="katex">
P.据/span>据/span>=据/span>0.据/span>, 然后据span class="katex">
P.据/span>是素数,存在一对一的同性恋据span class="katex">
F据/span>P.据/span>→据/span>F据/span>。在后一种情况下,如果据span class="katex">
F据/span>是有限的,然后它有据span class="katex">
P.据/span>K.据/span>一些正整数的元素据span class="katex">
K.据/span>。据/P.>据/B.lockquote>
第一步是表明这一点据span class="katex">
P.据/span>是素数。这很清楚:假设据span class="katex">
P.据/span>不是素质和写作据span class="katex">
P.据/span>=据/span>一种据/span>B.据/span>那据span class="katex">
一种据/span>那据/span>B.据/span>据据/span>P.据/span>。然后据span class="katex-display">
(据/span>一种据/span>⋅据/span>1据/span>)据/span>⋅据/span>(据/span>B.据/span>⋅据/span>1据/span>)据/span>=据/span>一种据/span>时代据/span>
(据/span>1据/span>+据/span>1据/span>+据/span>⋯据/span>+据/span>1据/span>)据/span>B.据/span>时代据/span>
(据/span>1据/span>+据/span>1据/span>+据/span>⋯据/span>+据/span>1据/span>)据/span>=据/span>一种据/span>B.据/span>时代据/span>
1据/span>+据/span>1据/span>+据/span>⋯据/span>+据/span>1据/span>=据/span>一种据/span>B.据/span>⋅据/span>1据/span>=据/span>P.据/span>⋅据/span>1据/span>=据/span>0.据/span>那据/span>由于这在一个领域,这意味着据span class="katex">
一种据/span>⋅据/span>1据/span>=据/span>0.据/span>或者据span class="katex">
B.据/span>⋅据/span>1据/span>=据/span>0.据/span>,违反了最小性据span class="katex">
P.据/span>。据/P.>据P.>接下来,定义同态据span class="katex">
F据/span>:据/span>F据/span>P.据/span>→据/span>F据/span>经过据span class="katex">
F据/span>(据/span>一种据/span>⋅据/span>1据/span>)据/span>=据/span>一种据/span>⋅据/span>1据/span>;这显然是一对一,显然是同性恋。所以有一份副本据span class="katex">
F据/span>P.据/span>里面据span class="katex">
F据/span>。据/P.>据P.>其余的声明从理论中遵循据一种href="//www.parkandroid.com/wiki/vector-spaces/" class="wiki_link" title="矢量空间GydF4y2Ba" target="_blank">矢量空间据/一种>:据span class="katex">
F据/span>是一个矢量空间据span class="katex">
F据/span>P.据/span>,并且由于它有限,因此它有一些维度据span class="katex">
K.据/span>和某种依据据span class="katex">
X据/span>1据/span>那据/span>......据/span>那据/span>X据/span>K.据/span>,这样的每个元素据span class="katex">
F据/span>可以在表格中独一无二的写作据span class="katex">
一种据/span>1据/span>X据/span>1据/span>+据/span>⋯据/span>+据/span>一种据/span>K.据/span>X据/span>K.据/span>为了据span class="katex">
一种据/span>一世据/span>∈据/span>F据/span>P.据/span>,所以有据span class="katex">
P.据/span>K.据/span>元素的选择据span class="katex">
F据/span>。据/P.>据!-- end-proof -->
有限田地的分类据/h2>
以上结果是有限字段分类的第一步。这个完整分类非常完全并且非常强大。部分证据将在下面的部分中给出。据/P.>据B.lockquote class="theorem">
(1)每个有限的领域都有据span class="katex">
P.据/span>K.据/span>一些主要的元素据span class="katex">
P.据/span>和一些积极的整数据span class="katex">
K.据/span>。据/P.>据P.>(2)对于每个素数据span class="katex">
P.据/span>和正整数据span class="katex">
K.据/span>,有一个有限的领域据span class="katex">
P.据/span>K.据/span>元素。据/P.>据P.>(3)相同大小的任何两个有限场都是同构。据/P.>据P.>所以:有一个有限的领域据span class="katex">
F据/span>P.据/span>K.据/span>和据span class="katex">
P.据/span>K.据/span>元素(达到同构)。据/P.>据!-- end-theorem -->
请注意(1)已在上面证明。证明(2)需要推价和存在关于不可缩短多项式的存在定理,并且(3)的证据迅速从(2)的证明。据/P.>据/D.一世v>
构建有限田地据/h2>
如上所述,有限的领域据span class="katex">
P.据/span>K.据/span>元素包含一份副本据span class="katex">
F据/span>P.据/span>。这些领域的结构从多项式环开始据span class="katex">
F据/span>P.据/span>[据/span>X据/span>]据/span>一种变量中的多项式与系数据span class="katex">
F据/span>P.据/span>。有限的领域据span class="katex">
P.据/span>K.据/span>元素将是一个合适的据一种href="//www.parkandroid.com/wiki/ring-theory/" class="wiki_link" title="商GydF4y2Ba" target="_blank">商据/一种>这个戒指。据/P.>据B.lockquote class="example">
多项式据span class="katex">
X据/span>2据/span>+据/span>X据/span>+据/span>1据/span>是不可挽回的据span class="katex">
F据/span>2据/span>[据/span>X据/span>]据/span>(实际上,它是唯一不可挽回的二次多项式据span class="katex">
F据/span>2据/span>[据/span>X据/span>]据/span>)。考虑商量据span class="katex-display">
F据/span>=据/span>F据/span>2据/span>[据/span>X据/span>]据/span>/据/span>(据/span>X据/span>2据/span>+据/span>X据/span>+据/span>1据/span>)据/span>。据/span>卖据span class="katex">
α.据/span>=据/span>X据/span>那据span class="katex">
F据/span>可以被认为是据span class="katex">
F据/span>2据/span>[据/span>α.据/span>]据/span>在哪里据span class="katex">
α.据/span>2据/span>+据/span>α.据/span>+据/span>1据/span>=据/span>0.据/span>。据/P.>据P.>然后是元素据span class="katex">
F据/span>是据span class="katex">
0.据/span>那据/span>1据/span>那据/span>α.据/span>那据/span>α.据/span>+据/span>1据/span>。多项式据span class="katex">
α.据/span>较大程度可以减少到线性多项式中据span class="katex">
α.据/span>通过使用关系据span class="katex">
α.据/span>2据/span>=据/span>α.据/span>+据/span>1据/span>。所以据span class="katex">
F据/span>有四个要素,实际上并不难检查这一点据span class="katex">
F据/span>是一个领域。特别是,据span class="katex">
α.据/span>(据/span>α.据/span>+据/span>1据/span>)据/span>=据/span>1据/span>, 所以据span class="katex">
α.据/span>和据span class="katex">
α.据/span>+据/span>1据/span>是乘法反转(和据span class="katex">
1据/span>是它自己的反向)。据/P.>据P.>这个字段表示为据span class="katex">
F据/span>4.据/span>。注意特征据span class="katex">
F据/span>4.据/span>是据span class="katex">
2据/span>。不要误会据span class="katex">
F据/span>4.据/span>对于整数模数据span class="katex">
4.据/span>: 例如,据span class="katex">
Z.据/span>/据/span>(据/span>4.据/span>)据/span>不是一个领域,而且据span class="katex">
1据/span>+据/span>1据/span>据/span>=据/span>0.据/span>在据span class="katex">
Z.据/span>/据/span>(据/span>4.据/span>)据/span>。据/P.>据!-- end-example -->
一般来说,如果据span class="katex">
K.据/span>是一个领域,据span class="katex">
K.据/span>[据/span>X据/span>]据/span>/据/span>(据/span>F据/span>(据/span>X据/span>)据/span>)据/span>如果且才有才能是一个字段据span class="katex">
F据/span>(据/span>X据/span>)据/span>是不可制定的。证据与证明相同据span class="katex">
Z.据/span>/据/span>(据/span>N据/span>)据/span>如果且仅当据span class="katex">
P.据/span>是素质:如果据span class="katex">
一种据/span>(据/span>X据/span>)据/span>是相对素质的据span class="katex">
F据/span>(据/span>X据/span>)据/span>,有一种形式据一种href="//www.parkandroid.com/wiki/bezouts-identity/" class="wiki_link" title="Bezout的身份GydF4y2Ba" target="_blank">Bezout的身份据/一种>产生多项式据span class="katex">
B.据/span>(据/span>X据/span>)据/span>那据/span>G据/span>(据/span>X据/span>)据/span>这样据span class="katex">
一种据/span>(据/span>X据/span>)据/span>B.据/span>(据/span>X据/span>)据/span>+据/span>F据/span>(据/span>X据/span>)据/span>G据/span>(据/span>X据/span>)据/span>=据/span>1据/span>。因为所有非零据span class="katex">
一种据/span>(据/span>X据/span>)据/span>谁程度小于据span class="katex">
F据/span>(据/span>X据/span>)据/span>是相对素质的据span class="katex">
F据/span>(据/span>X据/span>)据/span>, 因为据span class="katex">
F据/span>(据/span>X据/span>)据/span>是不可缩短的,这给了所有非零元素的乘法反转据span class="katex">
K.据/span>[据/span>X据/span>]据/span>/据/span>(据/span>F据/span>(据/span>X据/span>)据/span>)据/span>。据/span> 元素的数量据span class="katex">
F据/span>P.据/span>[据/span>X据/span>]据/span>/据/span>(据/span>F据/span>(据/span>X据/span>)据/span>)据/span>将会据span class="katex">
P.据/span>K.据/span>, 在哪里据span class="katex">
K.据/span>是的据span class="katex">
F据/span>(据/span>X据/span>)据/span>,因为这个字段的每个元素都可以唯一写作据span class="katex-display">
一种据/span>0.据/span>+据/span>一种据/span>1据/span>X据/span>+据/span>⋯据/span>+据/span>一种据/span>K.据/span>-据/span>1据/span>X据/span>K.据/span>-据/span>1据/span>那据/span>由于所有更高的力量据span class="katex">
X据/span>可以通过关系减少到较低的力量据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>。据/P.>据P.>所以这种结构给出了有限的大小据span class="katex">
P.据/span>K.据/span>适用于所有青睐据span class="katex">
P.据/span>和正整数据span class="katex">
K.据/span>,如果总有任何程度的不可缩短的多项式据span class="katex">
F据/span>P.据/span>[据/span>X据/span>]据/span>。这并不明显。据/P.>据/D.一世v>
存在不可缩短的多项式据/h2>
这是一个基本事实,在分类定理证明中是必不可少的:据/P.>据B.lockquote>
让据span class="katex"> P.据/span>是一个素质和据span class="katex"> N据/span>一个正整数。然后,在据span class="katex"> F据/span>P.据/span>[据/span>X据/span>]据/span>那据B.r> X据/span>P.据/span>N据/span>-据/span>X据/span>=据/span>D.据/span>|据/span>N据/span>π据/span>de据/span>(据/span>F据/span>)据/span>=据/span>D.据/span>F据/span>(据/span>X据/span>)据/span>蒙特雷雷。据/span>π据/span>F据/span>(据/span>X据/span>)据/span>。据/span>
例如,在据span class="katex"> F据/span>2据/span>[据/span>X据/span>]据/span>那据span class="katex-display"> X据/span>1据/span>6.据/span>-据/span>X据/span>=据/span>X据/span>(据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>2据/span>+据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>4.据/span>+据/span>X据/span>+据/span>1据/span>)据/span>(据/span>X据/span>4.据/span>+据/span>X据/span>3.据/span>+据/span>1据/span>)据/span>(据/span>X据/span>4.据/span>+据/span>X据/span>3.据/span>+据/span>X据/span>2据/span>+据/span>X据/span>+据/span>1据/span>)据/span>。据/span>
右侧是所有不可缩短的多项式的产物据span class="katex">
F据/span>2据/span>[据/span>X据/span>]据/span>学位据span class="katex">
1据/span>那据/span>2据/span>那据/span>或者据span class="katex">
4.据/span>。据/P.>据P.>据strong>证明事实:据/strong>
有几个步骤。首先要注意据span class="katex">
X据/span>P.据/span>N据/span>-据/span>X据/span>没有重复因素:如果它这样做了据一种href="//www.parkandroid.com/wiki/derivatives-of-polynomials-easy/" class="wiki_link" title="衍生物GydF4y2Ba" target="_blank">衍生物据/一种>将与它分享一个共同的因素,但它的衍生品就是据span class="katex">
-据/span>1据/span>,没有非凡的因素。据/P.>据P.>现在假设据span class="katex">
F据/span>(据/span>X据/span>)据/span>是不可挽回的程度据span class="katex">
D.据/span>|据/span>N据/span>。然后据span class="katex">
F据/span>P.据/span>[据/span>X据/span>]据/span>/据/span>(据/span>F据/span>(据/span>X据/span>)据/span>)据/span>是一个有限的订单领域据span class="katex">
P.据/span>D.据/span>。因此,如果据span class="katex">
F据/span>(据/span>X据/span>)据/span>据/span>=据/span>X据/span>,在那个领域据span class="katex">
X据/span>P.据/span>D.据/span>-据/span>1据/span>=据/span>1据/span>,通过概括据一种href="//www.parkandroid.com/wiki/fermats-little-theorem/" class="wiki_link" title="费玛的小定理GydF4y2Ba" target="_blank">费玛的小定理据/一种>。这意味着据span class="katex">
F据/span>(据/span>X据/span>)据/span>划分据span class="katex">
X据/span>P.据/span>D.据/span>-据/span>1据/span>-据/span>1据/span>。如果据span class="katex">
D.据/span>|据/span>N据/span>那据span class="katex">
(据/span>P.据/span>D.据/span>-据/span>1据/span>)据/span>|据/span>(据/span>P.据/span>N据/span>-据/span>1据/span>)据/span>, 所以据span class="katex">
F据/span>(据/span>X据/span>)据/span>划分据span class="katex">
X据/span>P.据/span>N据/span>-据/span>1据/span>-据/span>1据/span>那据/span>因此划分据span class="katex">
X据/span>P.据/span>N据/span>-据/span>X据/span>。据/P.>据P.>最后一步是表明没有其他因素划分据span class="katex">
X据/span>P.据/span>N据/span>-据/span>X据/span>。这是通过诱导和使用这一事实来完成的据span class="katex">
GCD.据/span>(据/span>P.据/span>K.据/span>-据/span>1据/span>那据/span>P.据/span>N据/span>-据/span>1据/span>)据/span>=据/span>P.据/span>GCD.据/span>(据/span>K.据/span>那据/span>N据/span>)据/span>-据/span>1据/span>,但省略了细节。据span class="katex">
□据/span>
采取双方的程度给出了一个有用的必然结果:据/P.>据B.lockquote>
推论:据/strong>
用于素数据span class="katex">
P.据/span>和正整数据span class="katex">
N据/span>, 让据span class="katex">
ν据/span>P.据/span>(据/span>N据/span>)据/span>是度量的黑色不可缩短多项式的数量据span class="katex">
N据/span>。然后据B.r>
P.据/span>N据/span>=据/span>D.据/span>|据/span>N据/span>σ.据/span>D.据/span>ν据/span>P.据/span>(据/span>D.据/span>)据/span>
和据一种href="//www.parkandroid.com/wiki/mobius-function/" class="wiki_link" title="Möbius反演GydF4y2Ba" target="_blank">Möbius反演据/一种>那据B.r>
ν据/span>P.据/span>(据/span>N据/span>)据/span>=据/span>N据/span>1据/span>D.据/span>|据/span>N据/span>σ.据/span>μ.据/span>(据/span>N据/span>/据/span>D.据/span>)据/span>P.据/span>D.据/span>。据/span>
从第二个公式看不难看出据span class="katex">
ν据/span>P.据/span>(据/span>N据/span>)据/span>总是严格大于据span class="katex">
0.据/span>。(这据span class="katex">
P.据/span>N据/span>总和的术语严格大于所有条款的所有其余部分的绝对值的总和。)因此总是有一个不可缩短的多项式据span class="katex">
F据/span>P.据/span>学位据span class="katex">
N据/span>,因此有限的大小领域据span class="katex">
P.据/span>N据/span>。据/P.>据B.lockquote class="example">
继续上面的例子,据span class="katex-display">
ν据/span>2据/span>(据/span>4.据/span>)据/span>=据/span>4.据/span>1据/span>(据/span>μ.据/span>(据/span>1据/span>)据/span>2据/span>4.据/span>+据/span>μ.据/span>(据/span>2据/span>)据/span>2据/span>2据/span>+据/span>μ.据/span>(据/span>4.据/span>)据/span>2据/span>1据/span>)据/span>=据/span>4.据/span>1据/span>(据/span>1据/span>6.据/span>-据/span>4.据/span>+据/span>0.据/span>)据/span>=据/span>3.据/span>。据/span>
让据span class="katex">
F据/span>2据/span>[据/span>X据/span>]据/span>是具有系数的多项式的环据span class="katex">
F据/span>2据/span>=据/span>Z.据/span>/据/span>2据/span>Z.据/span>。召回多项式是据strong>不可减少据/strong>如果它没有较小程度的不合适因素。据/P.>据P.>例如,有三个不可挽回的多项式据span class="katex">
4.据/span>在据span class="katex">
F据/span>2据/span>[据/span>X据/span>]据/span>,即据span class="katex-display">
X据/span>4.据/span>+据/span>X据/span>+据/span>1据/span>那据/span>X据/span>4.据/span>+据/span>X据/span>3.据/span>+据/span>1据/span>那据/span>X据/span>4.据/span>+据/span>X据/span>3.据/span>+据/span>X据/span>2据/span>+据/span>X据/span>+据/span>1据/span>。据/span> 有多少学位17个不可缩短的多项式据span class="katex">
F据/span>2据/span>[据/span>X据/span>]据/span>还是据/span>
提示:阅读有限公地wiki!据/h6>
分类定理证明据/h2>
遗体仍然是表明任何两个相同大小的有限场都是同构。所以假设据span class="katex"> F据/span>是一个有限的大小据span class="katex"> P.据/span>K.据/span>, 然后让据span class="katex"> F据/span>1据/span>=据/span>F据/span>P.据/span>[据/span>X据/span>]据/span>/据/span>(据/span>F据/span>(据/span>X据/span>)据/span>)据/span>在哪里据span class="katex"> F据/span>(据/span>X据/span>)据/span>是不可挽回的程度据span class="katex"> K.据/span>。表明这一点据span class="katex"> F据/span>是同构的据span class="katex"> F据/span>1据/span>,它足以表明有一个根源据span class="katex"> F据/span>(据/span>X据/span>)据/span>在据span class="katex"> F据/span>。(然后有一个非血管同性恋据span class="katex"> F据/span>1据/span>→据/span>F据/span>通过发送定义据span class="katex"> X据/span>到根,并且任何非活动均匀的田间是注射的(为什么?),并且两组相同尺寸之间的任何重新注射贴图都是基础的。)据/P.>据P.>但考虑多项式据span class="katex"> X据/span>P.据/span>K.据/span>-据/span>X据/span>在据span class="katex"> F据/span>;通过Fermat的概括定理,这种多项式分裂成线性因素据span class="katex"> F据/span>:据span class="katex-display"> X据/span>P.据/span>K.据/span>-据/span>X据/span>=据/span>α.据/span>∈据/span>F据/span>π据/span>(据/span>X据/span>-据/span>α.据/span>)据/span>和以来据span class="katex"> F据/span>(据/span>X据/span>)据/span>|据/span>(据/span>X据/span>P.据/span>K.据/span>-据/span>X据/span>)据/span>,它也必须分成线性因素据span class="katex"> F据/span>,事实上它有据span class="katex"> K.据/span>独特的根据span class="katex"> F据/span>。据span class="katex"> □据/span>
分类定理意味着“该领域据span class="katex"> F据/span>P.据/span>K.据/span>和据span class="katex"> P.据/span>K.据/span>元素“存在并且定义很好(达到同构)。据/P.>据/D.一世v>
子场据/h2>
子场的据span class="katex">
F据/span>P.据/span>K.据/span>考虑到上述讨论,易于分类。邀请感兴趣的读者提供证明。据/P.>据B.lockquote class="theorem">
F据/span>P.据/span>D.据/span>是一个子场据span class="katex">
F据/span>P.据/span>K.据/span>如果并且只有据span class="katex">
D.据/span>|据/span>K.据/span>。在这种情况下,据span class="katex">
F据/span>P.据/span>D.据/span>包括元素据span class="katex">
β据/span>∈据/span>F据/span>P.据/span>K.据/span>满意据span class="katex">
β据/span>P.据/span>D.据/span>=据/span>β据/span>。据/P.>据!-- end-theorem -->
从...开始据span class="katex">
F据/span>1据/span>6.据/span>=据/span>F据/span>2据/span>[据/span>X据/span>]据/span>/据/span>(据/span>X据/span>4.据/span>+据/span>X据/span>3.据/span>+据/span>X据/span>2据/span>+据/span>X据/span>+据/span>1据/span>)据/span>=据/span>F据/span>2据/span>[据/span>α.据/span>]据/span>。然后有一份副本据span class="katex">
F据/span>4.据/span>里面据span class="katex">
F据/span>1据/span>6.据/span>。它包括据span class="katex">
0.据/span>那据/span>1据/span>那据/span>α.据/span>3.据/span>+据/span>α.据/span>2据/span>那据/span>α.据/span>3.据/span>+据/span>α.据/span>2据/span>+据/span>1据/span>。验证这些是四根根部很简单据span class="katex">
X据/span>4.据/span>-据/span>X据/span>在据span class="katex">
F据/span>1据/span>6.据/span>。据!-- end-example -->
应用程序据/h2>
有限领域的一个应用正在建设中据一种href="//www.parkandroid.com/wiki/lfsr/?wiki_title=linear feedback shift registers" class="wiki_link new" title="线性反馈移位寄存器GydF4y2Ba" target="_blank" rel="nofollow">线性反馈移位寄存器据/一种>,这是由据一种href="//www.parkandroid.com/wiki/linear-recurrence-relations/" class="wiki_link" title="线性复发关系GydF4y2Ba" target="_blank">线性复发关系据/一种>在据span class="katex"> F据/span>问:据/span>。据/span>这个想法是,在有限场上对线性复发关系的解决方案序列是据em>定期据/em>,并且该期间与特征多项式有关据span class="katex"> F据/span>(据/span>X据/span>)据/span>复发,以及有限田间的算术据span class="katex"> F据/span>问:据/span>[据/span>X据/span>]据/span>/据/span>(据/span>F据/span>(据/span>X据/span>)据/span>)据/span>=据/span>F据/span>问:据/span>D.据/span>在哪里据span class="katex"> D.据/span>=据/span>de据/span>(据/span>F据/span>)据/span>。据/span>LFSRS用于加密(创建伪随机数发生器),电路测试,数字通信和许多其他区域。据/P.>据P.>在据一种href="//www.parkandroid.com/wiki/number-theory/" class="wiki_link" title="数字论GydF4y2Ba" target="_blank">数字论据/一种>,解决方案据一种href="//www.parkandroid.com/wiki/general-diophantine-equations-modular-arithmetic/" class="wiki_link" title="蒸番啶方程GydF4y2Ba" target="_blank">蒸番啶方程据/一种>通常通过减少方程式进行分析据span class="katex"> P.据/span>对于鼎盛时期据span class="katex"> P.据/span>。据/span>方程式解析据span class="katex"> F据/span>P.据/span>通常比分析更容易据span class="katex"> 问:据/span>。据/span>例如,曲线上的点的结构据span class="katex"> 问:据/span>难以确定一般,但在点上有相当精确的界限据span class="katex"> F据/span>问:据/span>对于某些特性的曲线:如果点数是据span class="katex"> N据/span>问:据/span>那据/span>然后据span class="katex-display"> |据/span>N据/span>问:据/span>-据/span>(据/span>问:据/span>+据/span>1据/span>)据/span>|据/span>≤.据/span>2据/span>G据/span>问:据/span> 那据/span>在哪里据span class="katex"> G据/span>是一个易于计算的数字,称为据strong>Genus.据/strong>曲线。这些都是着名的据strong>hasse-weil边界据/strong>,在1974年证明。注意,这意味着所有但是有限的据span class="katex"> 问:据/span>那据/span> N据/span>问:据/span>>据/span>0.据/span>那据/span>即有一个观点。据/P.>据P.>有限字段的算法广泛用于加密应用程序,包括据一种href="//www.parkandroid.com/wiki/elliptic-curves/" class="wiki_link" title="椭圆曲线GydF4y2Ba" target="_blank">椭圆曲线据/一种>用于加密电子数据的公钥加密和AES(高级加密标准),它使用算术据span class="katex"> F据/span>2据/span>5.据/span>6.据/span>。据/span>
问题加载......据/P.>据P.Class="note-text">注意加载......据/P.>据P.Class="set-text">设置加载......据/P.>据/D.一世v>