From 8d14c5b4146053771a2bedef100992eadeca71ed Mon Sep 17 00:00:00 2001
From: queue-miscreant <cubebert549@gmail.com>
Date: Sun, 3 Aug 2025 22:56:15 -0500
Subject: [PATCH] minor cleanups in finite-field.2

---
 posts/finite-field/2/index.qmd | 52 ++++++++++++++++++----------------
 1 file changed, 28 insertions(+), 24 deletions(-)

diff --git a/posts/finite-field/2/index.qmd b/posts/finite-field/2/index.qmd
index 0f73102..00e412f 100644
--- a/posts/finite-field/2/index.qmd
+++ b/posts/finite-field/2/index.qmd
@@ -400,7 +400,7 @@ fromIndices ns = columns (\(_, f) r -> f r) (\(c, _) -> Headed c) $
 fromIndices' = (singleton (Headed "m") head <>) . fromIndices
 
 -- Symbolic representation of a power of a companion matrix (in Markdown)
-compPowSymbolic "" m = "*f*((*C*)^*" ++ m ++ "*^)"
+compPowSymbolic "" m = "*f*(*C*^*" ++ m ++ "*^)"
 compPowSymbolic x  m = "*f*((*C*~*" ++ x ++ "*~)^*" ++ m ++ "*^)"
 
 -- Spans of a given color
@@ -430,8 +430,9 @@ The polynomial *u* = ~2~31 occurs at positions 3, 6, 9, and 12
 It follows that the roots of *u* are cyclic of order 5, so this polynomial is irreducible,
   but *not* primitive.
 
-Naturally, $\hat U(X)$ can be factored as powers of (*C*~*s*~)^3^.
-We can also factor it more naively as powers of *C*~*u*~. Either way, we get the same sequence.
+Naturally, *u* (or a polynomial isomorphic to it) can be factored as powers of (*C*~*s*~)^3^.
+We can also factor it more naively as powers of *C*~*u*~.
+Either way, we get the same sequence.
 
 :::: {layout-ncol="2" layout-valign="center"}
 ::: {}
@@ -680,8 +681,9 @@ leastDeg2Char3Poly = deg2Char3Polys !! 1
 
 colorDeg3Char3 = colorByEval 3 [(14, "red"), (17, "blue")]
 
-markdown $ markdownTable (fromIndices'[1..8]) [
-    "":map colorDeg3Char3 (charPolyPows 3 leastDeg2Char3Poly)
+markdown $ markdownTable (fromIndices' [1..8]) [
+    compPowSymbolic "14" "m":
+      map colorDeg3Char3 (charPolyPows 3 leastDeg2Char3Poly)
   ]
 ```
 
@@ -697,15 +699,30 @@ Note that ${}_3 16 \sim 121_x = x^2 + 2x + 1$ and ${}_3 13 \sim 111_x = x^2 + x
   both of which have factors over GF(3).
 
 
-Power Graphs
-------------
+Irreducible Graphs
+------------------
 
 We can study the interplay of primitives, irreducibles, and their powers by converting
   our sequences into (directed) graphs.
-Each node in the graph represents a characteristic polynomial that appears over the field;
-  call the one under consideration *a*.
-If the sequence of polynomials generated by *C*~*a*~ contains contains another polynomial *b*,
-  then there is an edge from *a* to *b*.
+Each node in the graph will represent an irreducible polynomial over the field.
+Call the one under consideration *a*.
+If the sequence of characteristic polynomials generated by powers of *C*~*a*~ contains
+  contains another polynomial *b*, then there is an edge from *a* to *b*.
+
+```{haskell}
+-- Convert a polynomial to the integer representing it in characteristic n
+asPolyNum n = evalPoly n . fmap (`mod` n)
+
+irreducibleGraph d n = concatMap (\(x:xs) -> map (x,) xs) polyKinClasses where
+  -- All irreducible polynomials of degree d in characteristic n
+  irrsOfDegree' = irrsOfDegree d n
+  -- Get "kin" polynomials as integers -- all those who appear as characteristic
+  -- polynomials in the powers of its companion matrix
+  getKinPolys = map (asPolyNum n . charpoly) . matrixPowersMod n . companion
+  -- Kin classes corresponding to each irreducible polynomial,
+  -- which is the first entry
+  polyKinClasses = map (nub . take (n^d) . getKinPolys) irrsOfDegree'
+```
 
 We can do this for every GF(*p*^*m*^).
 Let's start with the first few fields of characteristic 2.
@@ -728,19 +745,6 @@ Here are a few more of these graphs after doing so, over fields of other charact
 --| code-fold: true
 --| fig-cap: "GF(9)"
 
--- Convert a polynomial to the integer representing it in characteristic n
-asPolyNum n = evalPoly n . fmap (`mod` n)
-
-irreducibleGraph d n = concatMap (\(x:xs) -> map (x,) xs) polyKinClasses where
-  -- All irreducible polynomials of degree d in characteristic n
-  irrsOfDegree' = irrsOfDegree d n
-  -- Get "kin" polynomials as integers -- all those who appear as characteristic
-  -- polynomials in the powers of its companion matrix
-  getKinPolys = map (asPolyNum n . charpoly) . matrixPowersMod n . companion
-  -- Kin classes corresponding to each irreducible polynomial,
-  -- which is the first entry
-  polyKinClasses = map (nub . take (n^d) . getKinPolys) irrsOfDegree'
-
 -- Characteristic polynomial of the identity matrix
 eyePoly d n = asPolyNum n $ charpoly $ eye d
 -- Remove edges directed toward the characteristic polynomial of the identity