Difficult:
I can't follow any of the given proofs.
Reflective:
I think its interesting that all of these proofs come down to the fact that a sequence of integers that goes to zero in the limit must always be zero after some point.
Tuesday, December 10, 2013
Saturday, December 7, 2013
22.3-22.4, due on December 9
Difficult:
The instructions for calculating the Galois group of an arbitrary polynomial are absurd. You might as well crack RSA by factoring.
Reflective:
I was most interested in the concept of the discriminant. Rather than doing a difficult process to find out exactly what the Galois group is, just rely a simple one to find out if it is in the alternating group or not.
The instructions for calculating the Galois group of an arbitrary polynomial are absurd. You might as well crack RSA by factoring.
Reflective:
I was most interested in the concept of the discriminant. Rather than doing a difficult process to find out exactly what the Galois group is, just rely a simple one to find out if it is in the alternating group or not.
Tuesday, December 3, 2013
22.1-22.2, due on December 6
Difficult:
Learning the content wasn't so difficult, but I have no idea what motivated using phi and psi to calculate a Galois group.
Reflective:
I find it interesting that a result that is supposed to be important and far reaching, namely the fundamental theorem of Galois theory, relies on such a difficult operation, namely finding Galois groups of polynomials, to be applied directly.
Learning the content wasn't so difficult, but I have no idea what motivated using phi and psi to calculate a Galois group.
Reflective:
I find it interesting that a result that is supposed to be important and far reaching, namely the fundamental theorem of Galois theory, relies on such a difficult operation, namely finding Galois groups of polynomials, to be applied directly.
21.5-21.7, due on December 4
Difficult:
The proofs in sections 21.5-21.6 all fit together nicely, but as was mentioned in class, they don't really imply theorem 21.3
Reflective:
I find the table of cyclotomic polynomials in section 21.7 quite elegant. The fact that they all seem to have coefficients in the set {-1, 0, 1} is stated, but I find it interesting that they all seem to follow a simple pattern as well.
The proofs in sections 21.5-21.6 all fit together nicely, but as was mentioned in class, they don't really imply theorem 21.3
Reflective:
I find the table of cyclotomic polynomials in section 21.7 quite elegant. The fact that they all seem to have coefficients in the set {-1, 0, 1} is stated, but I find it interesting that they all seem to follow a simple pattern as well.
Sunday, December 1, 2013
21.3-21.4, due on December 2
Difficult:
I got lost at the end of the proof of theorem 21.2, most of page 243, specifically.
Reflective:
It's not exactly clear to me how this result is so close to proving Galois theory; I do remember needing roots of unity, but how it leads to finding the rest is beyond me. Nevertheless, it is interesting that the history of mathematics could have taken a different turn if Vandermonde was more rigorous in his assertion that roots of unity had non-trivial radical expressions.
I got lost at the end of the proof of theorem 21.2, most of page 243, specifically.
Reflective:
It's not exactly clear to me how this result is so close to proving Galois theory; I do remember needing roots of unity, but how it leads to finding the rest is beyond me. Nevertheless, it is interesting that the history of mathematics could have taken a different turn if Vandermonde was more rigorous in his assertion that roots of unity had non-trivial radical expressions.
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