#### DLO's II - uniqueness of rationals — 1 comment

: 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 — 2 comments

: One of the very first things we learn at higher math education is the different kinds of number sets … more →

#### Productive Ability — 4 comments

: The four kinds of ability are not equally distributed. Half of the population, fifty percent, are Gu … more →

#### Personality Typing

: 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

: I went out for a walk today, which is already a very unlikely event, but then I saw something that b … more →

:  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

: 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. — 9 comments

: ‘Rationals’, data and the wonders of analytics I think I have said this before and I wil … more →

#### Chapter 25: Half-Twist

: 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

: 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

: 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

: 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

: 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

: 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

: 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

: 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

: 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

: 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

: Prove that . Conclude that has degree 4 over . Find an irreducible polynomial satisfied by . The inc … more →

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