1:15 example and following example are a bit deceptive and confusing, but illustrate a point that was somewhat glossed over in the previous video about norms of Z[i] actually being a difference of squares with the polynomial variable being "squared by itself" independent of the coefficient: N(3+i) =(+3)^2 - (+1)^2(i)^2 =(+9) - (+1)(-1) =(+9) - (-1) =+9 +1 =+10
As described in the previous video, difference of squares came from a product of the polynomial with it's conjugate. It never stuck with me before how conjugate products, difference of squares, and norms are all part of "the same structural thing".
`ℚ[t]` is a ring of polynomials with rational coefficients in the variable `t`. So he's talking about units in that ring, which are the constant polynomials, `p(x) = 6` being one of them. Compare this to one of his previous videos in which he showed the multiplicative group of units modulo 12, `ℤ₁₂× = U₁₂`, which was the "clock" with just the odd hours: {1, 3, 5, 7, 9, 11}. A unit in a ring is each of the invertible elements in its multiplicative monoid. That is, every element of a ring that has a multiplicative inverse (so that when you multiply it with that inverse, you get `1`, the multiplicative identity). In a ring of polynomials, the only elements that have inverses are the constant polynomials, because when you try to invert anything else, you'll get something like `1/(t² + 3·t + 2)` which is no longer a polynomial (it would have to have a negative degree). So constant polynomials are the units in that ring. Compare this again to the example of the clock arithmetic modulo 12 he gave in one of his previous videos, and when he took only the odd numbers as the "(multiplicative) group of units", because these were the only elements that had multiplicative inverses. Even numbers didn't, because they shared a common factor of `2` with the modulus and were divisible by it.
@@bonbonpony Hi. The example he used was Z_8, not Z_12. And in Z_12, the group of units is {1,5,7,11}. 3 and 9 are zero divisors, not units. Also, note that all constant polynomials are units in Q[t], but not in Z[t]. See his example around 5:46
You allowed that there might be a factorization of 3+4i where the two factors had norms of 5 and 5 or possibly -5 and -5. In the gaussian integers norms are always positive so factoring 25 cannot have two numbers with norms of -5.
1:15 example and following example are a bit deceptive and confusing, but illustrate a point that was somewhat glossed over in the previous video about norms of Z[i] actually being a difference of squares with the polynomial variable being "squared by itself" independent of the coefficient:
N(3+i)
=(+3)^2 - (+1)^2(i)^2
=(+9) - (+1)(-1)
=(+9) - (-1)
=+9 +1
=+10
As described in the previous video, difference of squares came from a product of the polynomial with it's conjugate. It never stuck with me before how conjugate products, difference of squares, and norms are all part of "the same structural thing".
2:19 Whoa, negative norms!? Very cool 👍
Also, thanks for clearing up one of my misconceptions at 6:09!
luminary, really impressed.
5:29 why 6 is a unit in Q[t] ? If you calculate it's norm it's 36 ? I'm lost here
`ℚ[t]` is a ring of polynomials with rational coefficients in the variable `t`. So he's talking about units in that ring, which are the constant polynomials, `p(x) = 6` being one of them. Compare this to one of his previous videos in which he showed the multiplicative group of units modulo 12, `ℤ₁₂× = U₁₂`, which was the "clock" with just the odd hours: {1, 3, 5, 7, 9, 11}.
A unit in a ring is each of the invertible elements in its multiplicative monoid. That is, every element of a ring that has a multiplicative inverse (so that when you multiply it with that inverse, you get `1`, the multiplicative identity).
In a ring of polynomials, the only elements that have inverses are the constant polynomials, because when you try to invert anything else, you'll get something like `1/(t² + 3·t + 2)` which is no longer a polynomial (it would have to have a negative degree). So constant polynomials are the units in that ring.
Compare this again to the example of the clock arithmetic modulo 12 he gave in one of his previous videos, and when he took only the odd numbers as the "(multiplicative) group of units", because these were the only elements that had multiplicative inverses. Even numbers didn't, because they shared a common factor of `2` with the modulus and were divisible by it.
@@bonbonpony Hi. The example he used was Z_8, not Z_12. And in Z_12, the group of units is {1,5,7,11}. 3 and 9 are zero divisors, not units.
Also, note that all constant polynomials are units in Q[t], but not in Z[t]. See his example around 5:46
I think you should note that 3+4i factors into several different products.
(2+i) x(2+i)
(-1+2i) x (1-2i)
and
(-2-i) x (-2-i)
You allowed that there might be a factorization of 3+4i where the two factors had norms of 5 and 5 or possibly -5 and -5. In the gaussian integers norms are always positive so factoring 25 cannot have two numbers with norms of -5.
Thank you very helpful but please blink more often
It would be great if you could talk more slowly, it's very hard to understand anything because of that :/
YTube playback speed x0.75 is a thing
Have you tried putting it on x0.75 speed?