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become too independent and lax. Governments that must be responsive to
their citizens will tend to cycle back and forth between centralized and
decentralized profiles, whereas governments that are engaged in domination
and con
NED University of Engineering & Technology, Karachi
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58
GALOIS THEORY
of degree 3 over Z2; that is, both fields have 2 3 = 8 elements. By Moore's
theorem, both of these fields are isomorphic. More generally, one sees that
if f (x) and g(x) are irreducible polynomials over Z i, which have the same
degree, th
NED University of Engineering & Technology, Karachi
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50
GALOIS THEORY
Splitting Fields
Given a polynomial f(x) with coefficients in a field F, we are going to
describe the smallest field containing F and all the roots of f (x).
Definition. If F is a subfield of a field E, one also says that E is a field
ext
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GALOIS THEORY
Proof. Let
f (x) = g(x)h(x) = (bo + bix + + bmx m )(co + cix +  + ckx k );
by Theorem 39, we may assume that both g and h lie in Z[x]. By hypothesis, p I ao = boco so that p I bo or p I co, by Euclid's lemma in Z; since
p2 does not divid
NED University of Engineering & Technology, Karachi
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46
GALOIS THEORY
and
_q/360 2 y
= (1/(0 2 )z = wz.
We conclude that the roots of the cubic polynomial are given by the cubic
formula:
Z; CO
y
(0 2z ;
here
y 3 = ( r
(02 y an;
NrR)
and R = r 2 + 4q3 /27.
Example 16. If f (x) = x 3 15x 126, then f (x) is r
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GALOIS THEORY
Notice that each extension in this tower is pure and that E, the splitting field
of f (x), is contained in R'. Since F (w) F is a splitting field, Theorem 58
gives Gal(R' I F (co) 1 Gal(R'/F) and
Gal(k/F)/ Gal(R7F(w)
Gal(F(co)1 F).
Now Ga
NED University of Engineering & Technology, Karachi
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54
GALOIS THEORY
Proof. Let cfw_a1,. , a nd be a basis of EIB, and let I8 1 ,
, On be a basis
of B/F. It suffices to prove that Oj ai :1 <i < m,1 <f < n is a basis
of EIF.
This set spans E. If y E E, then there are bi in B with y = E bi ai .
But each bi
NED University of Engineering & Technology, Karachi
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GALOIS THEORY
Proof. If a is a primitive element of F, then every nonzero element of F
has the form ak for some integer k, and ak is a square if and only if k is
even. Since a and b are not squares, we have a = ak and b = am where
both k and m are odd.
NED University of Engineering & Technology, Karachi
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MECHANICAL ME201

Spring 2016
70
GALOIS THEORY
Proof. Consider the map Gal(E/F) > Z p of the theorem. If f (x) splits,
then Gal(E/F) = 1 and its image is trivial; if f (x) does not split, then
its image is a nontrivial subgroup of Z. But Z p has no proper nontrivial
subgroups, so that
NED University of Engineering & Technology, Karachi
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38
53.
GALOIS THEORY
(i) Prove that the zero ideal in a ring R is a prime ideal if and only if R
is a domain.
(ii) Prove that the zero ideal in a ring R is a maximal ideal if and only if
R is a field.
54. The ideal 1 in 7Z[x] consisting of all polynomials
NED University of Engineering & Technology, Karachi
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MECHANICAL ME201

Spring 2016
62
GALOIS THEORY
Proof. Define * : Gal(E/F) > Gal(B/F) by a 1* a113; Lemma 57
says that * does take its values in Gal(B/F). It is easily seen that * is
a homomorphism with ker * = Gal(E/B) [if a 1B = identity, then a is
an automorphism of E fixing 13],
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REFERENCES
[13] A. R. Magid, Lectures on Differential Galois Theory,
Mathematical Society, 1994.
American
[14] G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and Applications of Finite Groups, Dover, 1961, Originally published by Wiley,
191
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nonconformity to norms will lead to severe punishments, dynamics of the
emotions and the solidaritygenerating system intersect and mitigate the
vertical system of authority, the constant monitoring by superior officers, and
t
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tional attachments and solidarity among incumbents that revolve around
anticorporate unit sentiments and culture. This joint good thus aligns
incumbent subcultures and commitments against the normative expectations attached to
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6
The Dynamics of Organizations
highly moralized if defined by ideologies and values. Another incentive
system is coercive. Individuals perform activities in order to gain whatever
rewards are associated with incumbency and to avoid punishments for fa
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6
The Dynamics of Organizations
their costs, investments, and rewards in a fairly hardnosed way, which
generally does not cause the arousal of positive emotions that provide an
extra intrinsic reward associated with the production of a private good.
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Spring 2016
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