2021.10.13 MATH 3600 Finite Fields

52 days ago by calkin

a=mod(2,13) 
       
type(a) 
       
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>
a^10 
       
10
10
[[k,a^k] for k in srange(1,13)] 
       
[[1, 2],
 [2, 4],
 [3, 8],
 [4, 3],
 [5, 6],
 [6, 12],
 [7, 11],
 [8, 9],
 [9, 5],
 [10, 10],
 [11, 7],
 [12, 1]]
[[1, 2],
 [2, 4],
 [3, 8],
 [4, 3],
 [5, 6],
 [6, 12],
 [7, 11],
 [8, 9],
 [9, 5],
 [10, 10],
 [11, 7],
 [12, 1]]
a=mod(12,13) [[k,a^k] for k in srange(1,13)] 
       
[[1, 12],
 [2, 1],
 [3, 12],
 [4, 1],
 [5, 12],
 [6, 1],
 [7, 12],
 [8, 1],
 [9, 12],
 [10, 1],
 [11, 12],
 [12, 1]]
[[1, 12],
 [2, 1],
 [3, 12],
 [4, 1],
 [5, 12],
 [6, 1],
 [7, 12],
 [8, 1],
 [9, 12],
 [10, 1],
 [11, 12],
 [12, 1]]
13^2-1 
       
168
168
a=mod(5,13) alpha=sqrt(a) 
       
print(alpha) 
       
sqrt5
sqrt5
type(alpha) 
       
<type
'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>
<type 'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>
alpha^2 
       
5
5
type(alpha^2) 
       
<type
'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>
<type 'sage.rings.finite_rings.element_givaro.FiniteField_givaroElement'>
phi=(1+alpha)/2 print(phi) 
       
7*sqrt5 + 7
7*sqrt5 + 7
phi^168 
       
1
1
for i in srange(1,169): if phi^i==1: print(i) 
       
28
56
84
112
140
168
28
56
84
112
140
168
phibar=(1-alpha)/2 for i in srange(1,169): if phibar^i==1: print(i) 
       
28
56
84
112
140
168
28
56
84
112
140
168