Kategorien

## Every field extension of degree 2 is normal

Assertion Every field extension L:K of degree 2 is normal. Proof We show Let $$\alpha\in L\setminus K.$$ 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 […]

Kategorien

## A field extension of degree 2, that is not Galois

A field extension is Galois, if it’s normal and separable. Every field extension of degree 2 is normal. Proof We need to find a field extension of degree 2 that is not separable. is not separable, because the minimal polynomial $$y^2+x^2$$ of $$x$$ over $$\mathbb{Z}_2 (x^2)$$ has in $$\mathbb{Z}_2 (x)$$ the double zero $$x$$ because […]

Kategorien

## Q(i) and Q(√2) are isomorphic vector spaces, but not isomorphic fields

Assertion ℚ(i) and ℚ(√2) are isomorphic ℚ vector spaces. Proof i has the minimal polynomial X2 + 1 and √2 has the minimal polynomial X2 – 2. Both have the degree 2, so the elements are algebraic via ℚ. Thus ℚ(i) and ℚ(√2) are ℚ vector spaces of the same dimension 2 and thus isomorphic. […]