Blogs about: Rationals
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Finite extensions of the rationals contain only finitely many roots of unity
Let be a finite extension of . Prove that contains only finitely many roots of unity. Suppose to the contrary that contains infinitely many roots of unity. Now for each , there are only finitely many… more →
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DLO's II - uniqueness of rationals
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wrote 1 week ago: Now that we’ve proven that the rationals exist, we would like to know if it would make a diffe … more →
DLO's I - existence of the rationals
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wrote 2 weeks ago: One of the very first things we learn at higher math education is the different kinds of number sets … more →
Productive Ability
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wrote 1 month ago: The four kinds of ability are not equally distributed. Half of the population, fifty percent, are Gu … more →
Personality Typing
wrote 4 months ago: I’ve recently been very interested in personality typing. There are a couple of components, as … more →
I Saw Something That Got Me Thinking Today
wrote 5 months ago: I went out for a walk today, which is already a very unlikely event, but then I saw something that b … more →
The Rationals Deserve Some of Your Respect
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wrote 11 months ago: In 2010, 60′s garage-rock band The Rationals finally got some respect when they were voted i … more →
One Plus Everything That Is Not One
wrote 1 year ago: In the shower yesterday afternoon, where my mind is often prone to wander to mathematical curiositie … more →
Want to get value out of your data and analytics investment? Then deal with this issue before you buy the software.
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wrote 1 year ago: ‘Rationals’, data and the wonders of analytics I think I have said this before and I wil … more →
Chapter 25: Half-Twist
wrote 1 year ago: Lem led the robot to his home, accompanied by the rest of his group. The path they followed was conv … more →
Finite extensions of the rationals contain only finitely many roots of unity
wrote 1 year ago: Let be a finite extension of . Prove that contains only finitely many roots of unity. Suppose to the … more →
Compute the splitting field of x⁶-4 over QQ
wrote 1 year ago: Compute the splitting field of over and its degree. Note that factors as . (Using only the differenc … more →
Compute the splitting field of x⁴+x²+1 over QQ
wrote 1 year ago: Compute the splitting field of over , and its degree. Note that , where and . Evidently, the roots o … more →
Compute the splitting field of x⁴+2 over QQ
wrote 1 year ago: Compute the splitting field of over and its degree. The roots of exist in (if we don’t know th … more →
Compute the splitting field of x⁴-2 over QQ
wrote 1 year ago: Compute the splitting field of over , as well as its degree. Let denote the positive real fourth roo … more →
Argue that the regular 5-gon is constructible
wrote 1 year ago: Use the fact that satisfies the polynomial (to be proved later) to argue that the regular 5-gon is c … more →
Exhibit a quadratic field as a field of matrices
wrote 1 year ago: Let , where is a squarefree integer. Let be in , and consider the basis of over . Compute the matrix … more →
Show that a given family of extensions of the rationals does not contain a cube root of 2
wrote 1 year ago: Suppose , where for each . Prove that . We can see, by an inductive argument, that has degree over f … more →
Compute the degree of a given extension of the rationals
wrote 1 year ago: Compute the degrees of and over . Let . Evidently, . (WolframAlpha agrees.) That is, is a root of . … more →
Compute the degree of a given extension of the rationals
wrote 1 year ago: Prove that . Conclude that has degree 4 over . Find an irreducible polynomial satisfied by . The inc … more →
