Abstract algebra final

>abstract algebra final
>one of the questions depends on being able to find the roots of a 4th degree polynomial irreducible in Q
>no calculators
>we didn't study any methods for doing this in class
>spend a good 20 minutes guessing and checking, still don't get it
well there went 15 points down the drain

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What was the question? It's likely you somehow misunderstood what you were meant to do.

f(x) = x^4 - 2x^2 + 2
1. prove f(x) is irreducible over Q (easy)
2. find the zeroes of f(x)
3. If E is the splitting field of f(x) over Q, prove that [E:F] > 4
4. prove that Gal(E/F) is isomorphic to D4

The last two parts would have been easy if I just had the right roots, instead I had to bullshit my way through it with answers I knew were wrong, but hopefully I'll get some points for being somewhere in the ballpark in terms of method.

Lmao you brainlet, it's a quadratic in x^2

>x^4 - 2x^2 + 2
Bruh. Let u= x^2
Can I ask, how did you make it to an abstract algebra class before learning elementary algebra?

FUCK YOU IT'S NOT WHAT THE CLASS WAS ABOUT
THERE WAS NO REASON TO REQUIRE US FIND TO THE ROOTS, THIS IS FUCKING COMPUTER WORK

If someone gives me an answer for this, I will give you a high res pic of Alexandra Daddario's butt.

If I were the guy marking your exam I'd probably try and take as many marks away from you as I could get away with. You're the sort of idiot who should never have been let into university. I bet you actually have to study to pass your exams.

Dude, mathematics is about discovery. Even if you did not think of substitution, if you had at least tried to solve the equation p(x) = 0 you would have eventually figured out "Hey, I can complete the square!"

Anyways, I remember we did this stuff back in highschool. We were constantly given to solve shit like sin^2x + sinx + 1 = 0 or x^6 + x^3 + 1 = 0

Because of that I think your professor thought "Well, these students have been really nice to me. I'll give them one easy problem, Oh I know. Let's give them a blast from the past to see them how modern algebra connects to their high school days. Yeah, let's find a cute polynomil with complex roots. Yes, super easy. There you go students, with love."

kek I was fucking about trying to guess a root and factorising it down to a cubic or quadratic so I feel pretty dumb now, least I didn't do this on an exam.

RIP OP

youtube.com/watch?v=SgDZ9LhZ2_g

>This is what happens when an autist is given modern math to circle jerk about rigor and completely forgets the childlike spirit of problem solving

This is sad to witness actually.
>WAAAH WAAAH we did not do this exact problem in class before! I really do not want to use my brain! Come on, I memorized all the definitions and theorems and I can write rigorous proofs now, so why do I have to bother with learning how to solve problems? Who cares about solving problems? Not me!

holy shit literal brainlet right here... how did you get to abstract algebra in the first place? Change of variable is basic problem solving my dude...

FUCKING MORON if you get a problem handed to you in a controlled environment (like an exam) ALWAYS suspect there's some trick to solve it easily and without need of calculator.

>Anyways, I remember we did this stuff back in highschool
As did I, almost four years ago. Sorry I wasn't prepared to be tested on everything I ever studied today.

>problem asks what is x + x
>I needed to remember that 1+1=2
>I LEARNED THAT BACK IN PRE-SCHOOL I DIDN'T KNOW I WAS GONNA BE TESTED ON EVERYTHING I EVER STUDIED JESUS CHRIST, WHO CARES ABOUT 1+1=2 ANYWAY?

Dude, elementary math is something you never forget. Fun math problems are those that mix elementary math with modern techniques because those are the kinds of problems that don't need 6 months of intense research to solve. If you are the kind of math student who doesn't do putnams for fun, or goes to sites like brilliant.org to practice their high school math for fun then I don't know what to tell you.

Math is about discovery, having fun and being yourself. If you can't deal with that then the engineering department is to the left.

>so why do I have to bother with learning how to solve problems?
fuck professors who think "problem-solving" questions belong on exams
>nice work buddy you understand all the principles of the course sure hope you're clever enough to guess this trick before you run out of time hehe

But the only reason you learn math is to learn to solve more problems. If you learn higher math while forgetting elementary math then you are not even expanding the scope of the problems you can solve. Heck, higher math is actually very useless compared to elementary math. With elem. math you can solve billions of problems, while with each topic of higher math there are always like 2 or 3 special new problems you can solve.

I mean, when was the last time you solved an IMO by constructing a special topology on some set?

ITT people who'd waste 15 minutes of their test showing how they came up with the antiderivative of sin^2(x) on a post-calc 1 -course.

OP is right to be upset about the injustice he went through. Only brainlets waste their brain capacity to remember pointless arithmetic tricks at uni.

Why do people like you even study math? If your mind is not to keep a beautiful library of mathematical facts then just go get an engineering degree. Literally get out. You have no love for the art of math. It is not an injustice because you are supposed to be solving problems every day.

older, H0 = 70 km/s/pc (ly?)
if it's faster that means the universe has has more time to expand since the Hubble constant should be increasing over time (because the expansion of the universe is accelerating)
I guessed that, fuck I need to go study my Astronomy ASAP

>implying I have time to waste solving trivial problems any computer can solve in microseconds when I'm supposed to do rigorous research (i.e. actual math) to advance the field I'm working on

holy shit user
finding zeores is easy but anyone care to explain what irreducible over Q means?
wtf are all these words, splitting fields, gal(e/f), isomorphic to D4

Pointless abstract algebra shit no one cares about. It's just fancy words meant to disguise some stupidly trivial concepts so algebra brainlets can feel like they're doing something important.

Q is the set of rational numbers
that's all I know

>injustice
>pointless arithmetic tricks

HURR DURR I can't do a simple variable substitution

>stupidly trivial concepts

wow look at Mr Galois over here

Irreducible over Q is fancy for "there are no rational roots".

Splitting field of a polynomial is fancy for "Q but you are allowed to also put in the roots of the polynomial"

Gal(E/F) is the Galois group which just means the set of functions from E to itself that preserve elements of F.

D4 is just a group with 8 elements.

>durr I can solve roots me smart me memorized the quadratic formula at highschool

Can you even derive the quadratic formula?

>brainlet who can't solve a hs math problem
>doing research

But I can. I just type that shit in matlab and it's done. I'm efficient like that.

Jesus christ
t. CS nerd

Super weird, there always is a thread about some dude doing a course Im also currently doing but he already had to do the exam and he also is completely fucking stupid.
does anyone else gets this on Veeky Forums?

>Irreducible over Q is fancy for "there are no rational roots".
No, it means it can't be factored into other polynomials in Q[x]. That same polynomial times x still wouldn't have rational roots, but it would be reducible.

sry OP, just banter.
I also hate doing computations.

>That same polynomial times x still wouldn't have rational roots
>0 is not a rational number

Really gets that nogging jogging. With you and OP I think we have a whole team of algebra experts in Veeky Forums, I did not even know!

ah fuck. okay, x^2+1.

That's algebraists for you.

>too dumb for analysis
>too dumb for inverse problems
>too dumb for mathematical physics
>durr I wasted years of my life studying math so I can't drop out now... I know I'll study algebra!

But x^2 + 1 is irreducible on Q too.
You know, you were originally right in that irreducibility is a bit more sensitive than just not having rational roots because there are some special cases, but you have failed to construct an example.

Younger. Rough estimate of age of the universe is 1/H0 and the current best estimate is about 67 so 100 would mean younger

jfc, just complete the squares for ax^2+bx+c

>prove f(x) is irreducible in Q
>find zeroes of f(x)
you are supposed to use kronecker's field extension theorem to find the roots of f(x)

no amount of nogging jogging denies the fact your definition of irreducible is still trash

It was good enough. No one gives a shit about the special cases anyway.

>x^2 + 1 is irreducible on Q too.
Yeah, I meant multiply by x^2 + 1 instead, i.e. (x^2 + 1)(x^4 - 2x^2 + 2)

Dude last year I had a question on my one of my math finals. The question was about legendre polynomials using cos^2(x) but the question had 1 - sin^2(x) instead of cos. And I completely fucking blanked. That questions was worth 10% of my final grade in the class and I got a 0 on it.

The definition misses all linear polynomials in Q[x], which are irreducible, and catches all products of two polynomials that each have no rational roots, which are not irreducible.

using the square root formula for complex numbers, you can figure out that all 4 of the roots of x^4 - 2x^2 + 2 are +/- (sqrt(2)/2)* [sqrt(sqrt(2) + 1) +/- i*sqrt(sqrt(2) - 1)]. The smallest field containing these roots must contain sqrt(2), sqrt(sqrt(2) + 1), and, because sqrt(sqrt(2)+1) happens to be the multiplicative inverse of sqrt(sqrt(2) - 1), i will also be in the field, which means sqrt(sqrt(2) - 1) is in the field, too. Clearly, (because sqrt(sqrt(2)+1) and sqrt(sqrt(2) -1) are multiplicative inverses of each other), you will need 12 generating vectors to produce the splitting field (which is {1,sqrt(2)} X {1,i} X {1,sqrt(sqrt(2)+1),sqrt(sqrt(2)-1)} ). Obviously, Gal(E/F) is isomorphic to D_4 = D_2*(2) = K_4, as there are only 4 automorphisms, identity, swapping the sign on sqrt(2), swapping the sign on i, and doing both simultaneously.

Ha there is a quicker way to do it with a variable substitution. Whose the brainlet now? It's you.

>look at me I read a thing on Wikipedia who's the brainlet now?

You can solve it by dimensional analysis if you know fractions. Good luck

question two is piss easy, its literally grade 12 high-school math. just take the factors of the polynomial (aka the last number which is 2) and the factors you will have are -1,1,-2,2. now substitute the variable with these numbers. if a number gives youa zero. then use the factor that gave you zero and use synthetic division.

Not using quartic formula in 2017

stop bullying op

HAHAHAHAHAHAHAHAHAHAHAHAHA BRAINLET

>fuck professors who think "problem-solving" questions belong on exams
THE ABSOLUTE STATE OF EDUCATION

Plug in those numbers and see if you get zero. Try again

"waah I should be able to immediately solve any problem ever also without studying or applying myself" - this guy, probably