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# 数学代写|数论代写Number Theory代考|Index and Minimal Index of an Algebraic Number Field

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## 数学代写|数论代写Number Theory代考|Index and Minimal Index of an Algebraic Number Field

Let $K$ be an algebraic number field of degree $n$ over $\mathbb{Q}$. An element $\alpha \in O_K$ is called a generator of $K$ if $K=\mathbb{Q}(\alpha)$. By Theorem 6.4.3 $\alpha$ is a generator of $K$ if and only if $D(\alpha) \neq 0$. For a generator $\alpha$ of $K$, the index of $\alpha$ is the positive integer ind $\alpha$ given by
$$D(\alpha)=(\text { ind } \alpha)^2 d(K)$$
(see Definition 7.1.4). We now define the index $i(K)$ and minimal index $m(K)$ of the field $K$.
Definition 7.4.1 (Index of a field) The index of $K$ is
$$i(K)=\operatorname{gcd}{\text { ind } \alpha \mid \alpha \text { a generator of } K} .$$
Definition 7.4.2 (Minimal index of a field) The minimal index of $K$ is
$$m(K)=\min {\text { ind } \alpha \mid \alpha \text { a generator of } K} .$$
Clearly
$$i(K) \mid m(K) .$$

## 数学代写|数论代写Number Theory代考|Integral Basis of a Cyclotomic Field

Let $m$ be a positive integer. The number of positive integers less than or equal to $m$ that are coprime with $m$ is denoted by $\phi(m)$. The arithmetic function $\phi(m)$ is called Euler’s phi function. Let $\zeta_m$ be any primitive $m$ th root of unity. There are $\phi(m)$ primitive $m$ th roots of unity, namely $\zeta_m^r, r=1,2, \ldots, m,(r, m)=1$. Let $K_m=\mathbb{Q}\left(\zeta_m\right)$. It is easy to show that $K_m=\mathbb{Q}\left(\zeta_m^r\right)$ for any $r \in{1,2, \ldots, m}$ with $(r, m)=1$, so that $K_m$ is independent of the primitive $m$ th root of unity chosen. The field $K_m$ is called the $m$ th cyclotomic field. For odd $m$ the fields $K_m$ and $K_{2 m}$ coincide as $-\zeta_m$ is a primitive $2 m$ th root of unity. Clearly $\zeta_m$ is a root of the polynomial
$$f_m(x)=\prod_{\substack{r=1 \(r, m)=1}}^m\left(x-\zeta_m^r\right) .$$
It is known that $f_m(x) \in \mathbb{Z}[x]$ and that $f_m(x)$ is irreducible, so that
$$\operatorname{irr}_{\mathbb{Q}}\left(\zeta_m\right)=f_m(x)$$
Moreover, the degree of $f_m(x)$ is $\phi(m)$ so that
$$\left[K_m: \mathbb{Q}\right]=\phi(m) .$$
The smallest field containing both $K_m$ and $K_n$ is $K_{[m, n]}$, where $[m, n]$ denotes the least common multiple of $m$ and $n$. Also, $K_m \cap K_n=K_{(m, n)}$. If $m \not \equiv 2(\bmod 4)$ then $K_m \subseteq K_n$ holds if and only if $m \mid n$. Thus if $m$ and $n$ are distinct and not congruent to $2(\bmod 4)$ the cyclotomic fields $K_m$ and $K_n$ are distinct.
The next theorem gives an integral basis for $K_m$ as well as a formula for the discriminant $d\left(K_m\right)$.
Theorem 7.5.1 Let $m$ be a positive integer. Let $\zeta_m$ be a primitive mth root of unity. Let $K_m$ denote the cyclotomic field $\mathbb{Q}\left(\zeta_m\right)$. Then $\left{1, \zeta_m, \zeta_m^2, \ldots, \zeta_m^{\phi(m)-1}\right}$ is an integral basis for $K_m$. Further,
$$d\left(K_m\right)=(-1)^{\frac{\phi(m)}{2}} \frac{m^{\phi(m)}}{\prod_{p \mid m} p^{\frac{\phi(m)}{p-1}},}$$
where the product is over all primes $p$ dividing $m$.

## 数学代写|数论代写Number Theory代考|Index and Minimal Index of an Algebraic Number Field

$$D(\alpha)=(\text { ind } \alpha)^2 d(K)$$
(见定义7.1.4)。现在我们定义字段$K$的索引$i(K)$和最小索引$m(K)$。

$$i(K)=\operatorname{gcd}{\text { ind } \alpha \mid \alpha \text { a generator of } K} .$$

$$m(K)=\min {\text { ind } \alpha \mid \alpha \text { a generator of } K} .$$

$$i(K) \mid m(K) .$$

## 数学代写|数论代写Number Theory代考|Integral Basis of a Cyclotomic Field

$$f_m(x)=\prod_{\substack{r=1 (r, m)=1}}^m\left(x-\zeta_m^r\right) .$$

$$\operatorname{irr}{\mathbb{Q}}\left(\zeta_m\right)=f_m(x)$$ 此外，$f_m(x)$的程度是$\phi(m)$，所以 $$\left[K_m: \mathbb{Q}\right]=\phi(m) .$$ 同时包含$K_m$和$K_n$的最小字段是$K{[m, n]}$，其中$[m, n]$表示$m$和$n$的最小公倍数。还有，$K_m \cap K_n=K_{(m, n)}$。如果$m \not \equiv 2(\bmod 4)$，那么$K_m \subseteq K_n$当且仅当$m \mid n$成立。因此，如果$m$和$n$是不同的，而不等于$2(\bmod 4)$，那么切光场$K_m$和$K_n$是不同的。

$$d\left(K_m\right)=(-1)^{\frac{\phi(m)}{2}} \frac{m^{\phi(m)}}{\prod_{p \mid m} p^{\frac{\phi(m)}{p-1}},}$$

## MATLAB代写

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