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# 数学代写|数论代写Number Theory代考|MATH346 Norm and Trace

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## 数学代写|数论代写Number Theory代考|Norm and Trace

In this section, the notions of norm and trace ${ }^7$ are introduced and some important results related to these are proved which are used in the subsequent chapters.

Definition Let $K / F$ be a finite extension of fields, then $K$ is a finite-dimensional vector space over $F$. For $\alpha$ belonging to $K$, consider the $F$-linear transformation $T_\alpha$ of $K$ defined by $T_\alpha(\xi)=\alpha \xi$ for every $\xi \in K$. The characteristic polynomial of this linear transformation is called the characteristic polynomial of $\alpha$ relative to the extension $K / F$. Thus if $\left{v_1, v_2, \ldots, v_n\right}$ is a (vector space) basis of the extension $K / F$ and $\alpha v_i=\sum_{j=1}^n a_{i j} v_j, a_{i j} \in F$, then the characteristic polynomial of $\alpha$ relative to $K / F$ is determinant of the matrix $(X I-A)$, where $A=\left(a_{i j}\right)_{i, j}$ and $I$ is the $n \times n$ identity matrix.

Remark 1.12 With notations as in the above definition, it may be pointed out that the characteristic polynomial of $\alpha$ relative to $K / F$ is independent of the choice of the basis $\left{v_1, v_2, \ldots, v_n\right}$ of $K / F$. If $\left{v_1^{\prime}, v_2^{\prime}, \ldots, v_n^{\prime}\right}$ is another basis of $K / F$, then the matrix $B=\left(b_{i j}\right){i, j}$ of the linear transformation $T\alpha$ with respect to $\left{v_1^{\prime}, v_2^{\prime}, \ldots, v_n^{\prime}\right}$ defined by $\alpha v_i^{\prime}=\sum_{j=1}^n b_{i j} v_j^{\prime}$ is similar to the matrix $A$. In fact, $B=P A P^{-1}$, where $P$ is the transition matrix from $\left{v_1, v_2, \ldots, v_n\right}$ to $\left{v_1^{\prime}, v_2^{\prime}, \ldots, v_n^{\prime}\right}$, because
$$\left[\begin{array}{c} \alpha v_1 \ \alpha v_2 \ \vdots \ \alpha v_n \end{array}\right]=A\left[\begin{array}{c} v_1 \ v_2 \ \vdots \ v_n \end{array}\right],\left[\begin{array}{c} \alpha v_1^{\prime} \ \alpha v_2^{\prime} \ \vdots \ \alpha v_n^{\prime} \end{array}\right]=B\left[\begin{array}{c} v_1^{\prime} \ v_2^{\prime} \ \vdots \ v_n^{\prime} \end{array}\right],\left[\begin{array}{c} v_1^{\prime} \ v_2^{\prime} \ \vdots \ v_n^{\prime} \end{array}\right]=P\left[\begin{array}{c} v_1 \ v_2 \ \vdots \ v_n \end{array}\right]$$
and hence

$$\left[\begin{array}{c} \alpha v_1^{\prime} \ \alpha v_2^{\prime} \ \vdots \ \alpha v_n^{\prime} \end{array}\right]=P\left[\begin{array}{c} \alpha v_1 \ \alpha v_2 \ \vdots \ \alpha v_n \end{array}\right]=P A\left[\begin{array}{c} v_1 \ v_2 \ \vdots \ v_n \end{array}\right]=P A P^{-1}\left[\begin{array}{c} v_1^{\prime} \ v_2^{\prime} \ \vdots \ v_n^{\prime} \end{array}\right]$$
which shows that $B=P A P^{-1}$.

## 数学代写|数论代写Number Theory代考|Some Simple Properties of Norm and Trace

Let $K$ be an extension of degree $n$ of a field $F$. Let $\alpha, \beta$ be in $K$ and $a \in F$. Then the following hold:
(i) $\operatorname{Tr}{K / F}(a)=n a$ and $N{K / F}(a)=a^n$.
(ii) $\operatorname{Tr}{K / F}(\alpha+\beta)=\operatorname{Tr}{K / F}(\alpha)+\operatorname{Tr}{K / F}(\beta)$. (iii) $N{K / F}(\alpha \beta)=N_{K / F}(\alpha) N_{K / F}(\beta)$.
Proof The first two assertions follow immediately from the definition of norm and trace. We prove (iii). For an element $\alpha$ belonging to $K$, let $T_\alpha$ be as in the above definition and $M\left(T_\alpha\right)$ denote its matrix with respect to a fixed basis $\left{v_1, v_2, \ldots, v_n\right}$ of $K / F$. Note that $T_{\alpha \beta}=T_\alpha \circ T_\beta$. Therefore
$$M\left(T_{\alpha \beta}\right)=M\left(T_\alpha \circ T_\beta\right)=M\left(T_\beta\right) M\left(T_\alpha\right) .$$
Consequently
$$N_{K / F}(\alpha \beta)=\operatorname{det}\left(M\left(T_{\alpha \beta}\right)\right)=\operatorname{det}\left(M\left(T_\beta\right) M\left(T_\alpha\right)\right)=N_{K / F}(\alpha) N_{K / F}(\beta)$$
as desired.

## 数学代写|数论代写数论代考|范数与迹

.

$$\left[\begin{array}{c} \alpha v_1 \ \alpha v_2 \ \vdots \ \alpha v_n \end{array}\right]=A\left[\begin{array}{c} v_1 \ v_2 \ \vdots \ v_n \end{array}\right],\left[\begin{array}{c} \alpha v_1^{\prime} \ \alpha v_2^{\prime} \ \vdots \ \alpha v_n^{\prime} \end{array}\right]=B\left[\begin{array}{c} v_1^{\prime} \ v_2^{\prime} \ \vdots \ v_n^{\prime} \end{array}\right],\left[\begin{array}{c} v_1^{\prime} \ v_2^{\prime} \ \vdots \ v_n^{\prime} \end{array}\right]=P\left[\begin{array}{c} v_1 \ v_2 \ \vdots \ v_n \end{array}\right]$$
，因此

$$\left[\begin{array}{c} \alpha v_1^{\prime} \ \alpha v_2^{\prime} \ \vdots \ \alpha v_n^{\prime} \end{array}\right]=P\left[\begin{array}{c} \alpha v_1 \ \alpha v_2 \ \vdots \ \alpha v_n \end{array}\right]=P A\left[\begin{array}{c} v_1 \ v_2 \ \vdots \ v_n \end{array}\right]=P A P^{-1}\left[\begin{array}{c} v_1^{\prime} \ v_2^{\prime} \ \vdots \ v_n^{\prime} \end{array}\right]$$
，这表明$B=P A P^{-1}$ .

## 数学代写|数论代写数论代考| Norm和Trace的一些简单性质

.

(i) $\operatorname{Tr}{K / F}(a)=n a$和$N{K / F}(a)=a^n$ .
(ii) $\operatorname{Tr}{K / F}(\alpha+\beta)=\operatorname{Tr}{K / F}(\alpha)+\operatorname{Tr}{K / F}(\beta)$。(iii) $N{K / F}(\alpha \beta)=N_{K / F}(\alpha) N_{K / F}(\beta)$ .

$$M\left(T_{\alpha \beta}\right)=M\left(T_\alpha \circ T_\beta\right)=M\left(T_\beta\right) M\left(T_\alpha\right) .$$

$$N_{K / F}(\alpha \beta)=\operatorname{det}\left(M\left(T_{\alpha \beta}\right)\right)=\operatorname{det}\left(M\left(T_\beta\right) M\left(T_\alpha\right)\right)=N_{K / F}(\alpha) N_{K / F}(\beta)$$
as desired.

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