Challenge

[jMerliN]

If you can solve the problem in my signature adequately, I will teach you any programming language you want.

Good luck.

 

[Team Lion]

Is Q[x] simply the ring of rational numbers?

 

[jMerliN]

Is Q[x] simply the ring of rational numbers?

It's a polynomial ring.

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[Team Lion]

Sorry for the late reply, just recently had time to work on it.

I provide the steps for finding it as well.

Find a polynomial in Q[x] with 1+2^(1/3)+4^(1/3) as a root.

First I define x = 1 + 2^(1/3)+2^(2/3).
This problem equates to finding a polynomial P, such that P(x) = 0, with P having coefficients that are rational numbers but whose factors are either irrational or complex. One root is x, which is irrational, so more roots need to be found that are irrational or complex such that when multiplied together form a polynomial with rational coefficients, or a polynomial needs to be found simply that has a root of x with the required degree to make the polynomial have rational coefficients.

I shall solve the second case using linear algebra.

I compute x^n for numerous powers to find a basis for x^n
x^0 = 1
x^1 = 1 + 2^(1/3) + 2^(2/3)
x^2 = 5 + 4*2^(1/3) + 3*2^(2/3)
x^3 = 19 + 15*2^(1/3) + 12*2^(2/3)

By induction the basis then of x^n is [1 2^(1/3) 2^(2/3)]
Thus I conclude that the smallest such polynomial is of the degree 3, as there are 3 elements of the basis.

I rewrite each power as as a combination of the basis:
x^0 = [1 0 0]
x^1 = [1 1 1]
x^2 = [5 4 3]
x^3 = [19 15 12]

Now I solve for x^3 as a linear combination of the others.

[1 1 5 | 19]   [1 0 0 | 1]
[0 1 4 | 15] ~ [0 1 0 | 3]
[0 1 3 | 12]   [0 0 1 | 3]

And get x^3 = 1 + 3x + 3x^2

Or
x^3 - 3x^2 - 3x - 1

Which is a polynomial irreducible in Q[x]

 

[onemorejosh]

That's not irreducible. I can reduce it. I can reduce it wITH MY ROCKET LAUNCHER!@(!!( *BLAM*!(*%@#



---------- Post added at 12:43 PM ---------- Previous post was at 12:36 PM ----------

x^3 - 3x^2 - 3x - 1

x = 1 + cuberoot(2) + cuberoot(4)
x ~ 1 + 1.2599 + 1.5874
x ~ 3.8473

x^3 - 3x^2 - 3x - 1
~ (3.8473)^3 - 3(3.8473)^2 - 3(3.8473) - 1
~ 56.9466 - 44.4052 - 11.5419 - 1
~ 0

Looks good to me

 

[jMerliN]

Good job.

You didn't say why it's irreducible :O.

Not that it's hard to see why, but it's important that you don't just assume it, lol.

 

[Team Lion]

Its irreducible because if my polynomial is
f(x) = x^3-3x^2-3x-1
which is in Q[x], which is simply the group of all polynomial whose coefficients are rational numbers, I guarantee you their are no two polynomials g(x) and h(x) also in Q[x] such that
f(x) = g(x)h(x)

 

[jMerliN]

Its irreducible because if my polynomial is
f(x) = x^3-3x^2-3x-1
which is in Q[x], which is simply the group of all polynomial whose coefficients are rational numbers, I guarantee you their are no two polynomials g(x) and h(x) also in Q[x] such that
f(x) = g(x)h(x)

Since it's of degree 3, you only need to know that there are no linear factors, since all linear factors of a polynomial here are given by the rational root theorem, applying it shows that no linear factors exist, so it has to be irreducible.

 

[Team Lion]

Yeah. Abstract/Modern Algebra is a lot of fun. I remember the response a lot of people had when a professor I had in college said his research is in Algebra and people are like "Wow, I learned algebra in high school and this guy still doesn't know it." And some of these people were Mathematics majors.

 

[jMerliN]

Yeah. Abstract/Modern Algebra is a lot of fun. I remember the response a lot of people had when a professor I had in college said his research is in Algebra and people are like "Wow, I learned algebra in high school and this guy still doesn't know it." And some of these people were Mathematics majors.

I loved telling people "I'm taking Algebra" when they asked me what courses I was in, being a graduate Math student.

They're always like "wtf? algebra?? Didn't we take that in highschool?" And I would always say "no, you took a super simplified applied course of algebraic theory" and they'd say "ok..." and I'd just show them the text (D&F, what.. 1000 pages?) and they'd say "holy poop..." Always fun.

 

[Team Lion]

In Computer Engineering I always had problems with various computer scientists telling me how a certain processor works. They don't understand. I build them. I design them. I could draw out the logic circuitry for an n-bit processor. And they want to tell me how it works. They don't even know what a MUX is. Or how to draw out one using transistors. Or why your computer keeps 3.3 and 5V currents available and they need to be precise.

That was my favorite discussion. Most computer scientists aren't required to know how it works, and honestly, beyond assembler, they shouldn't need to.

(Of course I do know some computer scientists who do know a lot about the actual circuitry behind the processor, but they are very few amongst most of the computer scientists I've met.)

 

[SephX]

Thanks for the reading. This was the most interesting thread I have seen in a long time.

 

[jM2.me]

Thanks for the reading. This was the most interesting thread I have seen in a long time.

Even tho its 2 week bump, I agree. More of this stuff would be appreciated