LINEAR INDEPENDENCE OF CHARACTERS
LINEAR INDEPENDENCE OF CHARACTERS
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字符的线性独立性
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发表时间：
2008
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通讯作者：
Keith Conrad
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作者：
Keith Conrad
示例1.1。域同态K→F是一个特征标，它仅限于K的非零元素(即，使用G=K×)，并忽略域同态的加法方面。特别地，当L/K是域扩张时，AUT(L/K)的每个元素都是域同态L→L，因此是L×的特征标，取值于L×.示例1.2。对于所有的α∈F×，映射Z→F×by k 7→αk是Z的特征标。特征标G→F×可以被视为特殊函数G→F，但和不再是特征标(因为乘法映射的和通常不是可乘的，甚至可以取值0)。这个和只是一个函数G→F。函数G-→F在加法和F-尺度下形成一个向量空间。我们将证明不同的特征标G→F×与函数G→F线性无关。然后我们转向这种线性独立性的三个非常重要的应用：·正规基定理。·循环伽罗瓦扩张的希尔伯特定理90。·关于Kummer理论和Artin-Schreier理论的一些基本观点。我们将使用伽罗瓦理论来证明关于特征标的结果，但也可以使用特征标的线性无关性来证明伽罗瓦对应，就像在[3，14.2]和[9，4.2]中所做的那样。伽罗瓦理论的这种方法是由于Artin[1]，他在早期的治疗中“冒犯”了本原元素定理[7，第145页]。
Example 1.1. A field homomorphism K → F is a character by restricting it to the nonzero elements of K (that is, using G = K×) and ignoring the additive aspect of a field homomorphism. In particular, when L/K is a field extension every element of Aut(L/K) is a field homomorphism L→ L and therefore is a character of L× with values in L×. Example 1.2. For all α ∈ F×, the map Z→ F× by k 7→ αk is a character of Z. Characters G→ F× can be regarded as special functions G→ F and then can be added, but the sum is no longer a character (since the sum of multiplicative maps is usually not multiplicative, and could even take the value 0). The sum is just a function G → F . The functions G → F form a vector space under addition and F -scaling. We will prove that different characters G → F× are linearly independent as functions G → F . Then we turn to three very important applications of this linear independence: • The normal basis theorem. • Hilbert’s Theorem 90 for cyclic Galois extensions. • Some basic ideas in Kummer theory and Artin-Schreier theory. We will use Galois theory to prove results about characters, but linear independence of characters can also be used to prove the Galois correspondence, as done in [3, Sect. 14.2] and [9, Sect. 4.2] . This approach to Galois theory is due to Artin [1], who “took offense” [7, p. 145] at the role of the primitive element theorem in earlier treatments.