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Algebra Galoistheorie Mathematik

Every field extension of degree 2 is normal

Assertion

Every field extension L:K of degree 2 is normal.

Proof

We show \([L:K] = 2 \Rightarrow L\neq K\)

Let $$\alpha\in L\setminus K.$$

\(\alpha\) has at least degree 2 over K, that means the minimal polynomial $$m_{\alpha,K}$$ has at least degree 2.

By assumption $$m_{\alpha,K}$$ has at most degree 2. Therefore is $$L = K(\alpha)$$

Let $$\beta$$ be the second root of $$m_{\alpha,K}$$

$$-(\alpha + \beta)$$ is the coefficient of x in $$m_{\alpha,K}$$

Therefore is $$-(\alpha + \beta)\in K$$ and $$\beta\in K(\alpha)$$

$$\Rightarrow L$$ is the splitting field of $$m_{\alpha,K}$$ and therefore normal over K.

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