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# 数学代写|现代代数代考Modern Algebra代写|SEPARABILITY AND NORMALITY

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## 数学代写|现代代数代考Modern Algebra代写|SEPARABILITY AND NORMALITY

Recall from Section 46 that any polynomial $p(x)$ over a field $F$ has a splitting field over $F$, and any two splitting fields of $p(x)$ over $F$ are isomorphic. A polynomial of degree $n$ will have $n$ roots in a splitting field, but these $n$ roots need not be distinct (Problem 46.5). In Galois theory, irreducible polynomials create special problems if they have a repeated root in a splitting field. For this reason, we need the following definitions.

Definitions. A polynomial $p(x)$ of degree $n$ over a field $F$ is separable over $F$ if it has $n$ distinct roots in a splitting field $K$ over $F$. If $p(x)$ is not separable, it is inseparable. An algebraic element in an extension $K$ of $F$ is separable over $F$ if its minimum polynomial is separable over $F$. An algebraic extension $K$ of $F$ is a separable extension if every element of $K$ is separable over $F$.

The following theorem makes use of formal derivatives, which were introduced in Problem 34.13.

Theorem 47.1. A polynomial $p(x)$ over a field $F$ is separable over $F$ iff $p(x)$ and its formal derivative $p^{\prime}(x)$ are relatively prime in $K[x]$, where $K$ is a splitting field of $p(x)$; that is, iff they have no common factor of positive degree in $K[x]$.

## 数学代写|现代代数代考Modern Algebra代写|FUNDAMENTAL THEOREM OF GALOIS THEORY

Theorem 48.1 (Fundamental Theorem of Galois Theory). Assume that $E$ is a finite, separable, normal extension of a field $F$. Consider the correspondence defined by
$$K \rightarrow \operatorname{Gal}(E / K)$$
for each subfield $K$ of $E$ such that $K$ contains $F$. The correspondence (48.1) is one-to-one between the set of all subfields of $E$ that contain $F$ and the set of all subgroups of $\operatorname{Gal}(E / F)$. Moreover for each such $K$,
$$\begin{gathered} {[E: K]=|\operatorname{Gal}(E / K)|,} \ K=E_H \quad \text { for } \quad H=\operatorname{Gal}(E / K), \end{gathered}$$
and $K$ is normal over $F$ iff $\mathrm{Gal}(E / K)$ is a normal subgroup of $\operatorname{Gal}(E / F)$, in which case $\operatorname{Gal}(K / F) \approx \operatorname{Gal}(E / F) / \mathrm{Gal}(E / K)$.

PROOF. By Theorem 47.6 , the finite normal extension $E$ of $F$ is a splitting field of a polynomial $p(x)$ over $F$. If $F \subseteq K \subseteq E$, then $E$ is a splitting field of $p(x)$ over $K$. Also, $E$ is separable over $K$ by Theorem 47.3. Therefore, $[E: K]=|\mathrm{Gal}(E / K)|$ by Theorem 47.4, proving (48.2). By Theorem 47.5, if $H=\operatorname{Gal}(E / K)$ then $K=E_H$, proving (48.3).

We next prove that the correspondence in (48.1) is one-to-one and onto. To prove it is one-to-one, assume $K_1$ and $K_2$ intermediate subfields of $E$ and $F$, with $K_1 \neq K_2$. Then one of the subfields contains an element not in the other; assume $a \in K_1$ and $a \notin K_2$. If $G=\operatorname{Gal}\left(E / K_2\right)$, then $E_G=K_2$ by Theorem 47.5 , so there exists $\sigma \in G$ such that $\sigma(a) \neq a$. Thus $\sigma \notin \operatorname{Gal}\left(E / K_1\right)$ and $\operatorname{Gal}\left(E / K_1\right) \neq \operatorname{Gal}\left(E / K_2\right)$. Therefore the correspondence in (48.1) is one-to-one.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|FUNDAMENTAL THEOREM OF GALOIS THEORY

$$K \rightarrow \operatorname{Gal}(E / K)$$

$$\begin{gathered} {[E: K]=|\operatorname{Gal}(E / K)|,} \ K=E_H \quad \text { for } \quad H=\operatorname{Gal}(E / K), \end{gathered}$$
$K$比$F$正常，如果$\mathrm{Gal}(E / K)$是$\operatorname{Gal}(E / F)$的正常子组，在这种情况下$\operatorname{Gal}(K / F) \approx \operatorname{Gal}(E / F) / \mathrm{Gal}(E / K)$。

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## MATLAB代写

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