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
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Hi! Great video! Your my favorite RUclipsr 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!