Answered: Suppose that E is the splitting field…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Suppose that E is the splitting field of some polynomial over a field F
of characteristic 0. If Gal(E/F) is isomorphic to D₅, draw the subfield
lattice for the fields between E and F.

Expert Solution

Step 1

It is given that $E$ is the splitting field of some polynomial over a field $F$ of characteristic 0 and $G a l (E / F)$ is isomorphic to $D_{5}$ .

It is required to draw the sub-field lattice for the field between $E$ and $F$ .

it can be observed that,

$|D_{5}| = 2 \cdot 5 = 10$ .

So, the subgroup of $D_{5}$ has order 2 or 5 and it is a Sylow subgroup of $D_{5}$ .

Now, $D_{5}$ has 4 elements of order 5 (4 rotations) and 5 element of order 2 (the 5 reflections).

Then it can be observed that $D_{5}$ unique subgroup of order 5 and 5 sub group of order 2.

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