March 2015
Intermediate to advanced
344 pages
10h 18m
English
Content preview from Galois Theory, 4th Edition
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226 Finite Fields
19.9 By considering the field Z
2
(t), show that the Frobenius monomorphism is not
always an automorphism.
19.10* For which values of n does S
n
contain an element of order e(S
n
)?
(Hint: Use the cycle decomposition to estimate the maximum order of an el-
ement of S
n
, and compare this with an estimate of e(S
n
). You may need esti-
mates on the size of the nth prime: for example, ‘Bertrand’s Postulate’, which
states that the interval [n,2n] contains a prime for any integer n ≥ 1.)
19.11* Prove that in a finite field every element is a sum of two squares.
19.12 Mark the following true or false.
(a) There is a finite field with 124 elements.
(b) There is a finite field with 125 elements.
(c) There is a finite field with 126 elements.
(d) There is a finite ...
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