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Topic: Non-constructible numbers of degree 4
Replies: 8   Last Post: Dec 8, 2013 3:27 AM

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Jose Carlos Santos

Posts: 4,896
Registered: 12/4/04
Non-constructible numbers of degree 4
Posted: Dec 5, 2013 9:19 AM
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Hi all,

The standard way of proving that a given construction cannot be done
using compass and straightedge alone starting from the complex numbers 0
and 1 is to show that, if it could be done, then it would lead us to a
point _z_ such that there is no tower of fields

Q = K_0 <= K_1 <= K_2 <= ... <= K_n

with _z_ in K_n and such that the degree of each field over its
predecessor is 1 or 2. This shows that if _z_ transcendent or if it is
an algebraic number whose degree is no a power of 2, then _z_ is not
constructible. The most famous cases of impossible constructions which
lead to an algebraic number are the impossibility of trisecting a 60
degree angle and of doubling a cube; in each case, the standard proof
leads to an algebraic number of degree 3.

Can anyone tell me where to find examples of impossible constructions
which lead to algebraic numbers of degree four (or, more generally, some
power of 2)?

Best regards,

Jose Carlos Santos

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