︠8001f8ad-e248-4d41-83be-558028f39d53︠

︡1d643d4d-f148-44e8-9b58-07dd9322458e︡
︠e935e21c-a0d5-43b9-876c-8dcb1fc50493︠
2+3  # suma
︡4c30fcec-4baf-4d59-9aa2-545ba24c7b6a︡{"stdout":"5\n"}︡{"done":true}
︠90977827-c112-4611-99a3-89af0c4d00d3︠
7//3  # cociente en la división entera
︡bc666a47-c94b-406b-862d-77328ef43c0c︡{"stdout":"2\n"}︡{"done":true}
︠3cc4fd40-dc36-4568-9a6a-23215f584540︠
7%3   # resto de la división
︡9e4c69c0-d669-48ac-80a3-81cc40fcaac5︡{"stdout":"1\n"}︡{"done":true}
︠9132047a-d8f5-4048-a410-0f2985ec424b︠
factor(60) # factorización entera
︡3aa6b28b-00a7-4e2b-888e-f967b74499cd︡{"stdout":"2^2 * 3 * 5\n"}︡{"done":true}
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# Cuerpos finitos primos F_p
F = GF(7)          # Cuerpo primo F_7
︡f6245258-781a-4d38-8bb0-e6a30ddbbef8︡{"done":true}
︠1615b7b6-ef87-4791-9388-8506c2bb15b3s︠
F.list()           # Lista de elementos de F
︡9a8ddabb-f206-4c97-93b7-c1a1aabcc6ee︡{"stdout":"[0, 1, 2, 3, 4, 5, 6]\n"}︡{"done":true}
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a = F(3); b = F(5) # Elementos de F
︡372a4e6c-e6fa-4c84-aa3d-a93733bdc824︡{"done":true}
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a + b              # suma
a * b              # producto
a^-1               # inverso multiplicativo
︡85360394-e5d5-4669-bb46-bb3884b37755︡{"stdout":"1\n"}︡{"stdout":"1\n"}︡{"stdout":"5\n"}︡{"done":true}
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# Ejemplo: cuadrados en F_7
for x in F:
    print(x, x^2)
︡81bd2cbf-35c7-4c7a-afe9-7812c72c51c4︡{"stdout":"0 0\n1 1\n2 4\n3 2\n4 2\n5 4\n6 1\n"}︡{"done":true}
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# Polinomios sobre cuerpos finitos
R.<x> = PolynomialRing(GF(3))  # Anillo de polinomios sobre F_3
︡b1fc439c-202c-48c7-aee7-4df25cd9fb3e︡
︠efc039d6-4b10-4472-8ad3-27a2ea9ed35a︠
f = x^2 + 1
︡c62c2254-a353-4716-9d85-e340c7d05fab︡
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f.is_irreducible()             # ¿Es irreducible?
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f.factor()                     # Factorización
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# Extensiones de cuerpos finitos F_{p^m}
# Definir F_{2^4} usando polinomio irreducible x^4 + x + 1
R.<x> = PolynomialRing(GF(2))
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︠5bf15c79-7fc7-48ee-8244-409c2a001d52︠
F.<a> = GF(2^4, modulus=x^4 + x + 1)
︡885c1076-3125-42f4-853d-02e5d2787c0e︡
︠539fd510-7d8a-48f5-a4bb-870bc843df92︠
F.order()          # Cardinalidad del cuerpo (16)
︡fe0f52db-d9bf-4a2f-9386-108179cbda39︡
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F.multiplicative_generator()   # Generador del grupo multiplicativo
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# Potencias del generador
g = F.multiplicative_generator()
[g^i for i in range(1, F.order())]
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︠e292332f-de9e-4e6c-8d7e-aaaaf70198b0︠
# Operaciones en extensiones
alpha = a        # elemento del cuerpo extensión
alpha^5          # potencia
alpha.minpoly()  # polinomio mínimo de alpha
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# Más ejemplos
###############################################
# 1. Construir F_5. Calcular todos los inversos multiplicativos.
F5 = GF(5)
[(x, x^-1) for x in F5 if x != 0]
︡5288c7f0-43ad-41d2-ac10-042ab2c101b6︡
︠2c3aa265-e5f1-44dd-8d84-83384abba7c6︠
# 2. En F_7, verificar el pequeño teorema de Fermat
[(x, x^6) for x in GF(7)]
︡cd97f0aa-28ae-469b-a485-fc32c531e602︡
︠76196d63-535f-4f88-84aa-8f3ef09ec9f4︠
# 3. Construir F_{3^2} con polinomio irreducible de grado 2
R.<y> = PolynomialRing(GF(3))
F9.<b> = GF(3^2, modulus=y^2+1)
F9.list()
︡c7959a5a-eb9d-4b4d-b44d-210820bd13a4︡
︠2b92a8d6-2edd-4fc8-a931-e714b3078729︠

# 4. En F_{2^4}, buscar un generador del grupo multiplicativo
F.multiplicative_generator()