# 数学代写|伽罗瓦理论代写Galois Theory代考|MATH5020

## 数学代写|伽罗瓦理论代写Galois Theory代考|Primitive Element Theorem

Recall that a simple extension is one of the form $K(\theta) / K$. Many field extensions are presented in a manner that makes them appear not to be simple, but on further examination it turns out that they are. See Example 4.11(2), for instance.

We now give a classical criterion for a field extension in $\mathbb{C}$ to be simple. First, we need:
Lemma 6.12. If $p(t) \in K[t]$ is irreducible, it has no multiple zeros.
Proof. If $p(t)$ has a multiple zero $a$ then $p(t)=(t-a)^2 q(t)$ in $\mathbb{C}[t]$. The derivative
$$p^{\prime}(t)=2(t-a) q(t)+(t-a) q^{\prime}(t)$$
is divisible by $(t-a)$ in $\mathbb{C}[t]$, and the usual formula for the derivative of $t^n$ implies that $p^{\prime}(t) \in K[t]$. Therefore $p(t)$ and $p^{\prime}(t)$ have a common factor in $K[t]$ of degree at least 1 . But this is impossible since $\partial p^{\prime}=\partial p-1<\partial p$ and $p(t)$ is irreducible. Therefore $p(t)$ has no multiple zeros.

Using the degree, we can prove that every finite extension of a subfield of $\mathbb{C}$ is a simple extension, and conversely. A simpler and more general proof using Galois theory is given later in Theorem 17.29. The name of the theorem reflects classical terminology: if $L=K(\theta)$ then $\theta$ is a primitive element of $L$. The proof below comes from Brown (2010).

Theorem 6.13 (Primitive Element Theorem). Let $K, L$ be subfields of $\mathbb{C}$ with $[L: K]$ finite. Then there exists $\theta \in L$ such that $L=K(\theta)$.

Proof. By Lemma $6.11, L=K\left(\alpha_1, \ldots, \alpha_n\right)$ where the $\alpha_j$ are algebraic over $K$. The main point is to prove the case $n=2$ : if $\alpha_1, \alpha_2$ are algebraic over $K$ and $L=K\left(\alpha_1, \alpha_2\right)$, then there exists $\theta \in L$ such that $L=K(\theta)$. The theorem then follows by induction on $n$, since
$$L=K\left(\alpha_1, \ldots, \alpha_n\right)=K\left(\alpha_1, \alpha_2\right)\left(\alpha_3, \ldots, \alpha_n\right)=K\left(\theta, \alpha_3, \ldots, \alpha_n\right)$$
adjoining only $n-1$ algebraic elements to $K$.
To prove the result for $n=2$, write $\alpha_1=\alpha, \alpha_2=\beta$ to simplify notation. Consider an element
$$\gamma=\alpha+\lambda \beta$$
for $\lambda \in K$, and let $L=K(\alpha, \beta)$. We claim that $\gamma$ is a primitive element unless $\lambda$ is ‘bad’, that is, $\lambda$ belongs to a specific finite subset $S \subseteq K$, which we define during the proof; see $(6.4)$.

## 数学代写|伽罗瓦理论代写Galois Theory代考|Ruler-and-Compass Constructions

Already we are in a position to see some payoff. The degree of a field extension is a surprisingly powerful tool. Even before we get into Galois theory proper, we can apply the degree to a warm-up problem-indeed, several. The problems come from classical Greek geometry, and we will do something much more interesting and difficult than solving them. We will prove that no solutions exist, subject to certain technical conditions on the permitted methods.
According to Plato the only ‘perfect’ geometric figures are the straight line and the circle. In the most widely known parts of ancient Greek geometry, this belief had the effect of restricting the (conceptual) instruments available for performing geometric constructions to two: the ruler and the compass. The ruler, furthermore, was a single unmarked straight edge.

Strictly, the term should be ‘pair of compasses’, for the same reason we call a single cutting instrument a pair of scissors. However, ‘compass’ is shorter, and there is no serious danger of confusion with the navigational instrument that tells you which way is north. So ‘compass’ it is.

With these instruments alone it is possible to perform a wide range of constructions, as Euclid systematically set out in his Elements somewhere around 300 BC. This series of books opens with 23 definitions of basic objects ranging from points to parallels, five axioms (called ‘postulates’ in the translation hy Sir Thomas Heath), and five ‘common notions’ about equality and inequality. The first three axioms state that certain constructions may be performed:
(1) To draw a straight line from any point to any (distinct) point.
(2) To produce a finite straight line continuously in a straight line.
(3) To describe a circle with any centre and any distance.
The first two model the use of a ruler (or straightedge); the third models the use of a compass.

# 伽罗瓦理论代考

## 数学代写|伽罗瓦理论代写Galois Theory代考|Primitive Element Theorem

$$p^{\prime}(t)=2(t-a) q(t)+(t-a) q^{\prime}(t)$$

$$L=K\left(\alpha_1, \ldots, \alpha_n\right)=K\left(\alpha_1, \alpha_2\right)\left(\alpha_3, \ldots, \alpha_n\right)=K\left(\theta, \alpha_3, \ldots, \alpha_n\right)$$

$$\gamma=\alpha+\lambda \beta$$

## 数学代写|伽罗瓦理论代写Galois Theory代考|Ruler-and-Compass Constructions

(1) 从任意点到任意（不同的）点画一条直线。
(2) 在直线上连续产生一条有限直线。
(3) 以任意圆心和任意距离来描述一个圆。

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