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Show that the Galois group of a polynomial of degree n has orderdividing n!

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 8E: Show that every subgroup of an abelian group is normal.

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Question

Show that the Galois group of a polynomial of degree n has order
dividing n!

Expert Solution

Step 1

Let us consider that the polynomial,

$f (x) = \sum_{i = 0}^{n} c_{i} {x_{i}}^{n}, c_{i} \in F$

Also let $K$ be the splitting field of $f (x)$ over $F$ and $ϕ \in G a l (K / F)$ .

Let $S = \{a_{1}, a_{2}, \dots, a_{n}\}$ be the set of root of $f (x)$ then we see that $ϕ (a_{1}), ϕ (a_{2}), \dots, ϕ (a_{n})$ are distinct.

Let $l \leq m \leq n$ .

Since, $ϕ$ fixes $F$ , we can conclude that,

$\begin{array}{rcl} ϕ (f (a_{m})) & = & ϕ (\sum_{i = 0}^{l} c_{i} {a_{m}}^{i}) \\ = & \sum_{i = 0}^{l} c_{i} ϕ {(a_{m})}^{i} \end{array}$

Since, $a_{m}$ is root of $f (x)$ , so $f (m) = 0$ .

It follows that $f (ϕ (a_{m})) = 0$ .

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