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[parent] example of normal extension (Example)

Let $ F=\mathbb{Q}(\sqrt{2})$. Then the extension $ F/\mathbb{Q}$ is normal because $ F$ is clearly the splitting field of the polynomial $ f(x)=x^2-2$. Furthermore $ F/\mathbb{Q}$ is a Galois extension with $ \operatorname{Gal}(F/\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}$.

Now, let $ 2^{1/4}$ denote the positive real fourth root of $ 2$ and define $ K=F(2^{1/4})$. Then the extension $ K/F$ is normal because $ K$ is the splitting field of $ k(x)=x^2-\sqrt{2}$, and as before $ K/F$ is a Galois extension with $ \operatorname{Gal}(K/F)\cong \mathbb{Z}/2\mathbb{Z}$.

However, the extension $ K/\mathbb{Q}$ is neither normal nor Galois. Indeed, the polynomial $ g(x)=x^4-2$ has one root in $ K$ (actually two), namely $ 2^{1/4}$, and yet $ g(x)$ does not split in $ K$ into linear factors.

$\displaystyle g(x)=x^4-2=(x^2-\sqrt{2})\cdot(x^2+\sqrt{2})=(x-2^{1/4})\cdot(x+2^{1/4})\cdot(x^2+\sqrt{2})$

The Galois closure of $ K$ over $ \mathbb{Q}$ is $ L=\mathbb{Q}(2^{1/4},i)$.



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See Also: Galois extension, the compositum of a Galois extension and another extension is Galois, normal is not transitive, Galois is not transitive


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Cross-references: Galois closure, factors, root, real, positive, Galois extension, polynomial, splitting field, normal, extension
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This is version 1 of example of normal extension, born on 2004-07-30.
Object id is 6051, canonical name is ExampleOfNormalExtension.
Accessed 1995 times total.

Classification:
AMS MSC: 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
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