 #1
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Homework Statement
I am reading Dummit and Foote, Chapter 13  Field Theory.
I am currently studying Section 13.1 : Basic Theory of Field Extensions
I need some help with an aspect of Exercise 1 ... ...
Exercise 1 reads as follows:
Homework Equations
Relevant theorems for this exercise seem to be Theorems 4 and 6 of Section 13.1.
Theorem 4 reads as follows:
Theorem 6 reads as follows:
The Attempt at a Solution
##p(x) = x^2 + 9x + 6## is irreducible by Eisenstein ... ...
Now consider ##(x^2 + 9x + 10) = (x + 1) ( x^2  x + 10)##
and note that ##(x^2 + 9x + 10) = (x^2 + 9x + 6) + 4## ... ...
Now ##\theta## is a root of ##(x^2 + 9x + 6)## so that ...
##( \theta + 1) ( \theta^2  \theta +10) = ( \theta^2 + 9 \theta + 6) + 4 = 0 +4 = 4##
Thus ##( \theta + 1)^{1} = \frac{ ( \theta^2  \theta +10) }{4}## ... ...
Is that correct?
... BUT ... if it is correct I am most unsure of exactly where in the calculation we depend on ##p(x)## being irreducible ...
Can someone please explain where exactly in the above calculation we depend on ##p(x)## being irreducible?
Note that I am vaguely aware that we are calculating in ##\mathbb{Q} ( \theta )## ... which is isomorphic to ##\mathbb{Q} [x] / ( p(x) )## ... if ##p(x)## is irreducible ...
... BUT ...
I cannot specify the exact point(s) in the above calculation above where the calculation would break down if p(x) was not irreducible ... .. ... .. in fact, I cannot specify any specific points where the calculation would break down ... so I am not understanding the connection of the theory to this exercise ... ...
Can someone please help to clarify this issue ... ...
Peter
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