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Notes on 【Galois Theory】
02-28-2017, 02:54 PM
Post: #5
1.5 Splitting fields
1.5.1 Definition If $L/K$ is a field extension, we say that $f\in K[x]$ splits
(completely) over $L$ if $f = k\small (X-\alpha_1)\cdots(X-\alpha_n)$ where $k\in K,\;\alpha_j\in L$
$\,(j=\overline{1,n}).\quad L\,$ is called a splitting field for $f$ if $f$ fails to split over any
proper subfield of $L$, i.e. if $\small L = K(\alpha_1,\ldots,\alpha_n)$.

1.5.2 Remark Splitting fields always exist.
For if $g$ is an irreducible factor of $f$, then $\small K[X]/(g) = K(\alpha)$ is an extension
of $K$ for which $g(\alpha)=0\,(\alpha$ is the image of $X)$. The remainder theorem
implies that $g$(and hence $f$) splits off a linear factor. Induction implies that
there exists a splitting field $L$ for $f$, with $[L:K]\le n! \;(n = \deg(f))$ by
proposition 1.4.3.

Splitting fields are unique up to isomorphisms over $K$.

1.5.3 Proposition If $\theta:K\to K'$ is a field isomorphism with $f\in K[X]$]
corresponding to $g=\theta(f)\in K'[X]$. Then any splitting field $L$ of $f$ over
$K$ is isomorphic over $\theta$ to any splitting field $L'$ of $g$ over $K'$, and we have
the commutative diagram
$\qquad\qquad\qquad\qquad\qquad\begin{matrix}L & \overset{\tilde{\theta}}{\longrightarrow} & L'\\ \uparrow & & \uparrow\\ K & \overset{\theta}{\longrightarrow}& K'\end{matrix}$
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Messages In This Thread
Notes on 【Galois Theory】 - elim - 02-20-2017, 11:47 AM
1.3 Tests for irreducibility - elim - 02-27-2017, 11:18 PM
1.4 The degree of an extension - elim - 02-27-2017, 11:25 PM
1.5 Splitting fields - elim - 02-28-2017 02:54 PM

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