“dualble” → “dualable” (though to be fair this is not a real word so I guess it is subjective how you spell it)
“the book specify” → “the book doesn’t specify”
Regarding “one-hot”, it seems like the linear algebra concept that you want is “an element of the standard basis of R^d”. But it also seems that you are thinking about it in a combinatorial rather than linear algebra way, so I am not sure if you really want a linear algebra term for it.
Regarding the insolubility of the quintic, this is certainly an interesting result. But I’ve always thought it was a little odd to phrase this as that there is no “exact formula” for the root of a quintic polynomial. The theorem is just that there is no formula of a certain type (i.e. one built out of the standard four arithmetic operations plus integer roots). By analogy, there is no exact formula for the root of x^2 = 2 if by “exact formula” you mean an expression that just uses the standard four arithmetic operations. It seems to me that the fact that we can efficiently estimate the square root of 2 to arbitrary accuracy is exactly why it it is reasonable to say that we understand the solution. But the root of an integer polynomial can be estimated to arbitrary accuracy just as efficiently as a square root can.
Regarding insolubility of the quintic, I made a top level post with essentially the same point, because it deserves to be common knowledge, in full generality.
Typos:
“dualble” → “dualable” (though to be fair this is not a real word so I guess it is subjective how you spell it)
“the book specify” → “the book doesn’t specify”
Regarding “one-hot”, it seems like the linear algebra concept that you want is “an element of the standard basis of R^d”. But it also seems that you are thinking about it in a combinatorial rather than linear algebra way, so I am not sure if you really want a linear algebra term for it.
Regarding the insolubility of the quintic, this is certainly an interesting result. But I’ve always thought it was a little odd to phrase this as that there is no “exact formula” for the root of a quintic polynomial. The theorem is just that there is no formula of a certain type (i.e. one built out of the standard four arithmetic operations plus integer roots). By analogy, there is no exact formula for the root of x^2 = 2 if by “exact formula” you mean an expression that just uses the standard four arithmetic operations. It seems to me that the fact that we can efficiently estimate the square root of 2 to arbitrary accuracy is exactly why it it is reasonable to say that we understand the solution. But the root of an integer polynomial can be estimated to arbitrary accuracy just as efficiently as a square root can.
Regarding insolubility of the quintic, I made a top level post with essentially the same point, because it deserves to be common knowledge, in full generality.
Regarding “dualble”, I meant that entirely as a phonetic pun.
Thanks for the explanation of the quintic result! I suppose that’s slightly less mystic than my initial understanding.