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│ SageMath version 7.4, Release Date: 2016-10-18                     │
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sage: IntegerModRing(2)
Ring of integers modulo 2
sage: R1 = IntegerModRing(2)
sage: R1['x']
Univariate Polynomial Ring in x over Ring of integers modulo 2 (using NTL)
sage: R1['x,y,z']
Multivariate Polynomial Ring in x, y, z over Ring of integers modulo 2
sage: R2 = R1['x']
sage: R2.gens()
(x,)
sage: R2.gen()
x
sage: x = R2.gen()
sage: R2.ideal(1+x+x^2)
Principal ideal (x^2 + x + 1) of Univariate Polynomial Ring in x over Ring of integers modulo 2 (using NTL)
sage: I1 = R2.ideal(1+x+x^2)
sage: R2.quotient(I1)
Univariate Quotient Polynomial Ring in xbar over Ring of integers modulo 2 with modulus x^2 + x + 1
sage: R3 = R2.quotient(I1)
sage: R3.gens()
(xbar,)
sage: one = R3.one()
sage: one
1
sage: 1
1
sage: one == 1
True
sage: 1.parent()
Integer Ring
sage: one.parent()
Univariate Quotient Polynomial Ring in xbar over Ring of integers modulo 2 with modulus x^2 + x + 1
sage: R3.gens()
(xbar,)
sage: xbar = R3.gen()
sage: [[y*z for y in [0,one,xbar,one+xbar]] for z in [0,one,xbar,
....: one+xbar]]

[[0, 0, 0, 0],
 [0, 1, xbar, xbar + 1],
 [0, xbar, xbar + 1, 1],
 [0, xbar + 1, 1, xbar]]
sage: [[y+z for y in [0,one,xbar,one+xbar]] for z in [0,one,xbar,
....: one+xbar]]

[[0, 1, xbar, xbar + 1],
 [1, 0, xbar + 1, xbar],
 [xbar, xbar + 1, 0, 1],
 [xbar + 1, xbar, 1, 0]]
sage: I1.is_maximal()
True
sage: R3.is_field()
True
sage: R4 = RR['x']
sage: R4
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: R4 = QQ['x']
sage: R4
Univariate Polynomial Ring in x over Rational Field
sage: R4 = RR['x']
sage: CC
Complex Field with 53 bits of precision
sage: R4
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
sage: x = R4.gen()
sage: R4.quotient(R4.ideal(1+x^2))
Univariate Quotient Polynomial Ring in xbar over Real Field with 53 bits of precision with modulus x^2 + 1.00000000000000
sage: R5 = R4.quotient(R4.ideal(1+x^2))
sage: R5.is_field()
True
sage: xbar = R5.gen()
sage: (3+2*xbar)*(3/13-2/13*xbar)
1.00000000000000
sage: _.parent()
Univariate Quotient Polynomial Ring in xbar over Real Field with 53 bits of precision with modulus x^2 + 1.00000000000000
sage: (3+2*xbar)*(3/13-2/13*xbar)==R5.one()
True