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+# module 'zmod'
+
+# Compute properties of mathematical "fields" formed by taking
+# Z/n (the whole numbers modulo some whole number n) and an 
+# irreducible polynomial (i.e., a polynomial with only complex zeros),
+# e.g., Z/5 and X**2 + 2.
+#
+# The field is formed by taking all possible linear combinations of
+# a set of d base vectors (where d is the degree of the polynomial).
+#
+# Note that this procedure doesn't yield a field for all combinations
+# of n and p: it may well be that some numbers have more than one
+# inverse and others have none.  This is what we check.
+#
+# Remember that a field is a ring where each element has an inverse.
+# A ring has commutative addition and multiplication, a zero and a one:
+# 0*x = x*0 = 0, 0+x = x+0 = x, 1*x = x*1 = x.  Also, the distributive
+# property holds: a*(b+c) = a*b + b*c.
+# (XXX I forget if this is an axiom or follows from the rules.)
+
+import poly
+
+
+# Example N and polynomial
+
+N = 5
+P = poly.plus(poly.one(0, 2), poly.one(2, 1)) # 2 + x**2
+
+
+# Return x modulo y.  Returns >= 0 even if x < 0.
+
+def mod(x, y):
+	return divmod(x, y)[1]
+
+
+# Normalize a polynomial modulo n and modulo p.
+
+def norm(a, n, p):
+	a = poly.modulo(a, p)
+	a = a[:]
+	for i in range(len(a)): a[i] = mod(a[i], n)
+	a = poly.normalize(a)
+	return a
+
+
+# Make a list of all n^d elements of the proposed field.
+
+def make_all(mat):
+	all = []
+	for row in mat:
+		for a in row:
+			all.append(a)
+	return all
+
+def make_elements(n, d):
+	if d = 0: return [poly.one(0, 0)]
+	sub = make_elements(n, d-1)
+	all = []
+	for a in sub:
+		for i in range(n):
+			all.append(poly.plus(a, poly.one(d-1, i)))
+	return all
+
+def make_inv(all, n, p):
+	x = poly.one(1, 1)
+	inv = []
+	for a in all:
+		inv.append(norm(poly.times(a, x), n, p))
+	return inv
+
+def checkfield(n, p):
+	all = make_elements(n, len(p)-1)
+	inv = make_inv(all, n, p)
+	all1 = all[:]
+	inv1 = inv[:]
+	all1.sort()
+	inv1.sort()
+	if all1 = inv1: print 'BINGO!'
+	else:
+		print 'Sorry:', n, p
+		print all
+		print inv
+
+def rj(s, width):
+	if type(s) <> type(''): s = `s`
+	n = len(s)
+	if n >= width: return s
+	return ' '*(width - n) + s
+
+def lj(s, width):
+	if type(s) <> type(''): s = `s`
+	n = len(s)
+	if n >= width: return s
+	return s + ' '*(width - n)