Yes. This case of lattice refers to a partially ordered set.. with exactly what properties I don't remember.. they should definitely have joins and meets (a join of two elements a, b is the _maximum_ element (which is hypothesized to exist) of the subset { c : c ≤ a, c ≤ b }-that's the _intersection_ in case of a powerset; the meet is the dual concept) . Some properties such as associativity, commutativity, idempotence of joins and meets I think just follow. What doesn't follow in general is the distributivity w.r.t. each other and isn't hypothesized. Thus, the main theorem of galois theory needs a little more work showing that in the correspondence, these operations (dualized) correspond too-but that's not hard. There's of course another notion which also goes by the name of lattice: viz. a finitely generated Z-submodule (additive subgroup) of some R^n (R meaning the reals). This is not the one he was talking about, but this too shows up in these contexts, eg. when you're trying to compute a galois group (starting from an equation), dealing with factorization of rational polynomials, and in particular, the LLL algorithm. 👍
Sorry, if two lattices (in the first sense) are shown to be isomorphic as posets, the join and meet operations obviously correspond. No extra work required.
If K is a subfield of L, then L is a vector space over K. In the mentioned example, any (finite dimensional) vector space over F_8 must have cardinality 2^{3k}. In general F_p^{m} is a subfield of F_p^{n} iff m|n
Thank you so much for all these videos!!!
I am the 1337th viewer and the 69th like. I am so gonna ace my algebra exams haha.
I failed
@@Brien831 don't give up bro
I feel you, I'm on my way to the same fate 😢😭
Is "lattice" a mathematical terminology that can be precisely defined?
Yes. This case of lattice refers to a partially ordered set.. with exactly what properties I don't remember.. they should definitely have joins and meets (a join of two elements a, b is the _maximum_ element (which is hypothesized to exist) of the subset { c : c ≤ a, c ≤ b }-that's the _intersection_ in case of a powerset; the meet is the dual concept) . Some properties such as associativity, commutativity, idempotence of joins and meets I think just follow. What doesn't follow in general is the distributivity w.r.t. each other and isn't hypothesized. Thus, the main theorem of galois theory needs a little more work showing that in the correspondence, these operations (dualized) correspond too-but that's not hard.
There's of course another notion which also goes by the name of lattice: viz. a finitely generated Z-submodule (additive subgroup) of some R^n (R meaning the reals). This is not the one he was talking about, but this too shows up in these contexts, eg. when you're trying to compute a galois group (starting from an equation), dealing with factorization of rational polynomials, and in particular, the LLL algorithm. 👍
Sorry, if two lattices (in the first sense) are shown to be isomorphic as posets, the join and meet operations obviously correspond. No extra work required.
yes
11 minute mark: why is F8 not a subfield of F16?
Didnu figure it out?
@@jacobfertleman1980 not yet
If K is a subfield of L, then L is a vector space over K.
In the mentioned example, any (finite dimensional) vector space over F_8 must have cardinality 2^{3k}.
In general F_p^{m} is a subfield of F_p^{n} iff m|n
@@abderrahmaneprofmaths659 Thank you very much!
I have no idea what any of this means but I wanna know more!
ye