Mathematical Notation

"""
We can call each of these sets with Sage using the 
following commands. Comments are the result of the
input.
"""
CC
# Complex Field with 53 bits of precision
RR
# Real Field with 53 bits of precision
ZZ
# Integer Ring
QQ
# Rational Field
NN
# Non negative integer semiring
Zmod(11) # or `Integers(11)` or `IntegerModRing(11)` 
# Ring of integers modulo 11

"""
Example of defining a field and then its 
algebraic closure
"""
GF(3)
# Finite Field of size 3 , where GF stands for Galois Field 
GF(3).algebraic_closure()
# Algebraic closure of Finite Field of size 3

"""
If you want to find which field an element belongs to you can use the 
`.parent()` function
"""

x = 7
print(x.parent())
# Integer Ring

y = 3.5
print(y.parent())
# Real Field with 53 bits of precision

"""
If you want to "lift" an element from a quotient ring R/I to the ring R
use the `.lift()` function
"""
R = ZZ
RI = Zmod(11)
x =  RI(5)

print(x.parent())
# Ring of integers modulo 11

y = x.lift()
print(y.parent())
# Integer Ring

print(y in R)
# True