Math puzzles, part 1/3

[hennes]

Two days ago someone posted these links on an IRC channel:

1. http://bunbunmaru.com/wakaba/photos/src/1317203606.90948521.jpg
   
2. http://bunbunmaru.com/wakaba/photos/src/1317205463.5224561.png
   
3. http://bunbunmaru.com/wakaba/photos/src/1317205481.52651961.png

These contain links to three pictures with mathematical movie puzzles

Comments

[561]

Z/2Z is the ring of integers modulo 2; since 2 is prime this is actually a finite field. Either way, it is not just a set - that notation implies some sort of algebraic structure to a mathematician.

M_n(F_q) is the ring (Is it a ring, or only a group? Not sure, there will be some sort of algebraic structure there because F_q has structure) of n by n matrices with elements in the finite field F_q.

Z[\frac{1+\sqrt{-19}}{2}] is an extension ring...
http://mathworld.wolfram.com/ExtensionRing.html
Not sure whether the particular element \frac{1+\sqrt{-19}}{2} is important in this... the sqrt(-19) means this is going to be a complex number so the ring extension will be a subset of the complex numbers with finite degree (degree 2?) and hence algebraic. Anyone know if there is anything significant about that particular number?

[561]

http://planetmath.org/encyclopedia/MatrixGroup.html
M_n(F_q) is definitely a ring.

I would say that might well have three rings here... but the first I don't naturally look at and see ring (because it is a field and that has more structure than a ring). I mean, why not say Z/nZ if you meant ring?

So, field, matrix ring and extension ring? It is possible that M_n(F_q) has more structure than that of a matrix ring since F_q is not just a ring but a field but I would have to think to work that out. ;)

[561]

Apparently F_q being a field means that M_n(F_q)'s full structure is that of a Frobenius algebra...
http://en.wikipedia.org/wiki/Frobenius_algebra#Examples

This is supposed to be something to do with a film?

Giant algebra's attack queen of the castle ;)

[561]

I wonder if it is supposed to be Lord of the Rings? Obviously, if you are British then you would not think this because British Lords don't wear crowns so much... Do they in other countries or could it be a misconception that they do here?!