That's very kind! I'm glad you find the videos so helpful. 😁 Thanks for the suggestion - I am pretty sure I figured out the answer so maybe you'll see a video of it soon.
Abstract Algebra (particularly Galois Theory) is such an interesting topic to me. I gained a love for mathematics about a year ago in 10th grade, and later found out about Modern Algebra, and now I’ve become obsessed lol. I love just messing around with polynomials and learning about all the secrets they hold. A truly fascinating subject.
Thanks for great videos! More or less my level. If you have time to kill some day, you could show the splitting field of x⁴ - 20x² + 80 = 0. I think I know but not sure.
Of course! I'm always grateful to hear from people who enjoy my videos. When I started I wasn't sure if anyone would want to find the splitting field of this or that polynomial, so it's great to hear that people are enjoying it! For the polynomial you suggested, I will try to think about what the Galois group is. I try to do the splitting field and the Galois group in one video so maybe I'll upload something like that soon.
Reason I asked this particular question: cos(pi/20) = √(8+2√(10+2√5))/4 and so I wondered if 8+2√(10+2√5) could be factored (probably not but I can't prove it). To know that, I need to know what field to play in and it would also be convenient to have an equivalent of the "norm" of integers in that field (as well as a definition of "integer").@@coconutmath4928
Interesting. I don't know that much about number fields of higher degree. The extension Q(sqrt(10+2sqrt(5))) is an extension of finite degree, so it does have a ring of integers. If you wanted to find the norm you would need to find all the ways the field can be embedded into C and then multiply all of the embeddings of sqrt(10+2sqrt(5)) together. I'm not sure if this is enough for what you're trying to do but it was my main thought for how you could get started...
Hey thanks! I tried to reason that this polynomial is irreducible over Q because it is irreducible over Z, and that is because it has no roots in mod 2 therefore irreducible mod 2. Is this valid? Could you tell me where I went wrong?
Of course! The issue with this argument is that the polynomial could still factor as a product of two quadratics. And indeed it does: x^4+x^2+1 is equal to (x^2+x+1)(x^2+x+1) modulo 2, since any factorization of the polynomial over Z will still be valid if we reduce the coefficients mod 2. In general, not having any roots means that a polynomial is irreducible as long as it has degree 2 or 3, which is because there's a limit on how many ways the polynomial could factor -- once you start considering higher degrees, the existence or nonexistence of roots isn't enough anymore. So your approach is legit as long as the polynomial has degree
Hi! Great video! Your my favorite UA-camr for abstract algebra :) Video suggestion:
Find the splitting field of
18x^12 + 72x^8 + 54x^4 + 216.
That's very kind! I'm glad you find the videos so helpful. 😁 Thanks for the suggestion - I am pretty sure I figured out the answer so maybe you'll see a video of it soon.
Abstract Algebra (particularly Galois Theory) is such an interesting topic to me. I gained a love for mathematics about a year ago in 10th grade, and later found out about Modern Algebra, and now I’ve become obsessed lol.
I love just messing around with polynomials and learning about all the secrets they hold.
A truly fascinating subject.
Thanks for great videos! More or less my level. If you have time to kill some day, you could show the splitting field of x⁴ - 20x² + 80 = 0. I think I know but not sure.
Of course! I'm always grateful to hear from people who enjoy my videos. When I started I wasn't sure if anyone would want to find the splitting field of this or that polynomial, so it's great to hear that people are enjoying it!
For the polynomial you suggested, I will try to think about what the Galois group is. I try to do the splitting field and the Galois group in one video so maybe I'll upload something like that soon.
Reason I asked this particular question: cos(pi/20) = √(8+2√(10+2√5))/4 and so I wondered if 8+2√(10+2√5) could be factored (probably not but I can't prove it). To know that, I need to know what field to play in and it would also be convenient to have an equivalent of the "norm" of integers in that field (as well as a definition of "integer").@@coconutmath4928
Interesting. I don't know that much about number fields of higher degree. The extension Q(sqrt(10+2sqrt(5))) is an extension of finite degree, so it does have a ring of integers. If you wanted to find the norm you would need to find all the ways the field can be embedded into C and then multiply all of the embeddings of sqrt(10+2sqrt(5)) together.
I'm not sure if this is enough for what you're trying to do but it was my main thought for how you could get started...
Hey thanks! I tried to reason that this polynomial is irreducible over Q because it is irreducible over Z, and that is because it has no roots in mod 2 therefore irreducible mod 2. Is this valid? Could you tell me where I went wrong?
Of course! The issue with this argument is that the polynomial could still factor as a product of two quadratics. And indeed it does: x^4+x^2+1 is equal to (x^2+x+1)(x^2+x+1) modulo 2, since any factorization of the polynomial over Z will still be valid if we reduce the coefficients mod 2.
In general, not having any roots means that a polynomial is irreducible as long as it has degree 2 or 3, which is because there's a limit on how many ways the polynomial could factor -- once you start considering higher degrees, the existence or nonexistence of roots isn't enough anymore. So your approach is legit as long as the polynomial has degree
Such a clear explanation, thank you!