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What are these things called? Can someone explain what the undirected edges represent?

Example Diagram 1

Definitions: $F_{4096}$, $F_{64}$, $F_{16}$, $F_8$, $F_4$, and $\mathbb{Z}/2\mathbb{Z}$ are all finite fields.

First example

Example Diagram 2

Definitions:

  • $k$ is a field and $k[x]$ is the set of polynomials whose coefficients are in $k$.
  • $K$ is a field and $K^{\prime} = k[x]/(f(x))$, where $f(x)\in k[x]$ is irreducible.
  • $\phi : k[x] \to K^{\prime}$ is a homomorphism.

I assume the directed edges with the $\phi$ label just represent a homomorphism.

Second example

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The first looks like a Hasse Diagram. The second looks like a Commutative Diagram. –  user2468 Apr 6 '12 at 3:40
    
Ops. I did not scroll down the page; An answer was already posted! –  user2468 Apr 6 '12 at 3:43

1 Answer 1

up vote 2 down vote accepted

The first picture is the lattice of subfields of $\mathbb{F}_{2^{12}}$ (or technically, the diagram of it, as Mariano points out). The unlabelled edges represent inclusion as sets (one $\subset$ the other). See also.

The second is a commutative diagram, and $\phi$ a quotient map. The unlabelled edges might represent an evaluation map (replacing instances of the formal variable $x$ with a concrete element $\in k$.)

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(The Hasse diagram of the latttice of subfields, I'd say) –  Mariano Suárez-Alvarez Apr 6 '12 at 3:29
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(Actually, the long vertical line going from the bottom to the top is weird... and kills Hasseness) –  Mariano Suárez-Alvarez Apr 6 '12 at 3:34

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