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Galois Theory, 4th Edition
book

Galois Theory, 4th Edition

by Ian Nicholas Stewart
March 2015
Intermediate to advanced content levelIntermediate to advanced
344 pages
10h 18m
English
Chapman and Hall/CRC
Content preview from Galois Theory, 4th Edition
68 Field Extensions
Q(i,i,
5,
5). As written, it appears to require the adjunction of four new elements.
Clearly just two, i and
5, suffice. But we claim that in fact only one element is
needed, because L = L
0
where L
0
= Q(i +
5), which is obviously simple. To prove
this, it is enough to show that i L
0
and
5 L
0
, because these imply that L L
0
and L
0
L, so L = L
0
. Now L
0
contains
(i +
5)
2
= 1 + 2i
5 + 5 = 4 + 2i
5
Thus it also contains
(i +
5)(4 + 2i
5) = 14i 2
5
Therefore it contains
14i 2
5 + 2(i +
5) = 16i
so it contains i. But then it also contains (i +
5) i =
5. Therefore L = L
0
as
claimed, and the extension Q(i,i,
5,
5) : Q is in fact simple. ...
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Publisher Resources

ISBN: 9781482245837