abstract algebra II
damn, this is the most exciting math I've done since 5th grade. I took algebra last year and we barely did anything exciting, well at least when we were defining groups and rings and modules etc. Maybe there were a few exciting things from algebra 1 I remember-- like you can check whether a polynomial is irreducible by modding the coefficient field out by a prime (thus obtaining a finite # of roots to check), for example 8x^3 - 6x -1 mod out by 5 so you get the same poly over Z/5Z -> 3x^3 -x -1 and then check roots {0,1,2,3,4}. you get -1, 1, 1, 2, 2 respectively for their evaluations, i.e. all nonzero, so no roots over Z/5Z --> no roots over Z --> no roots over Q.
anyhow, I've only been to 3 classes of algebra 2, unfortunately we already had to do an 8 proof pset, but to make up for it we've done at least one exciting thing! from the theory of field extensions, we proved that there is no comprehensive way to trisect an angle using straightedge & compass. to construct any angle x, you need to be able to construct lengths cosx sinx to define it. the field containing any point constructed with s.e. & compass lies in a chain of quadratic field extensions of the rationals, so any polynomial that cosx or sinx satisfies must have degree 2^n. For some x's, for example 20 (degrees, not radians), the minimal (i.e. irreducible) polynomial that cosx (equivalently sinx) satisfies has a fucked up degree, for ex. in 20's case 3 not = 2^n, so it is unconstructible! so you can't trisect 60 degrees! and same theory states why you can't construct a regular 7-gon!
now for all practical purposes this is bullshit. you can get arbitrarily close to constructing 20 degrees since you can construct the whole of Q the rationals which is dense -> you can get arbitrarily close to cosx. but it's still cool. and more exciting than most things I do, like sleeping, eating, fucking, and corporate finance.
anyhow, I've only been to 3 classes of algebra 2, unfortunately we already had to do an 8 proof pset, but to make up for it we've done at least one exciting thing! from the theory of field extensions, we proved that there is no comprehensive way to trisect an angle using straightedge & compass. to construct any angle x, you need to be able to construct lengths cosx sinx to define it. the field containing any point constructed with s.e. & compass lies in a chain of quadratic field extensions of the rationals, so any polynomial that cosx or sinx satisfies must have degree 2^n. For some x's, for example 20 (degrees, not radians), the minimal (i.e. irreducible) polynomial that cosx (equivalently sinx) satisfies has a fucked up degree, for ex. in 20's case 3 not = 2^n, so it is unconstructible! so you can't trisect 60 degrees! and same theory states why you can't construct a regular 7-gon!
now for all practical purposes this is bullshit. you can get arbitrarily close to constructing 20 degrees since you can construct the whole of Q the rationals which is dense -> you can get arbitrarily close to cosx. but it's still cool. and more exciting than most things I do, like sleeping, eating, fucking, and corporate finance.