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Rings





In [1]:



    


RRing = Rings()



    











In [2]:



    


[method for method in dir(RRing) if method[0] is not r'_']



    


















    
Out[2]:








['AdditiveAssociative',
 'AdditiveCommutative',
 'AdditiveInverse',
 'AdditiveUnital',
 'Algebras',
 'Associative',
 'CartesianProducts',
 'Commutative',
 'Distributive',
 'Division',
 'ElementMethods',
 'Endsets',
 'Facade',
 'Facades',
 'Finite',
 'FinitelyGenerated',
 'FinitelyGeneratedAsMagma',
 'Homsets',
 'Infinite',
 'Inverse',
 'IsomorphicObjects',
 'Metric',
 'NoZeroDivisors',
 'ParentMethods',
 'Quotients',
 'Realizations',
 'SubcategoryMethods',
 'Subobjects',
 'Subquotients',
 'Topological',
 'Unital',
 'WithRealizations',
 'additional_structure',
 'all_super_categories',
 'an_instance',
 'axioms',
 'base_category',
 'category',
 'category_graph',
 'db',
 'dump',
 'dumps',
 'element_class',
 'example',
 'extra_super_categories',
 'full_super_categories',
 'hom_category',
 'is_abelian',
 'is_full_subcategory',
 'is_subcategory',
 'join',
 'meet',
 'morphism_class',
 'or_subcategory',
 'parent',
 'parent_class',
 'rename',
 'required_methods',
 'reset_name',
 'save',
 'structure',
 'subcategory_class',
 'super_categories',
 'version']

















In [3]:



    


RRing.axioms()



    


















    
Out[3]:








frozenset({'AdditiveAssociative',
           'AdditiveCommutative',
           'AdditiveInverse',
           'AdditiveUnital',
           'Associative',
           'Distributive',
           'Unital'})

















In [4]:



    


RCommutRing = CommutativeRing( RR ) # a "base" ring for input is needed



    











In [5]:



    


RCommutRing.is_commutative()



    


















    
Out[5]:








True

Abstract Algebra: Theory and Applications, Section on Rings has some useful examples:





In [6]:



    


ZZRing = ZZ()
RRRing = RR()
CCRing = CC()
QQRing = QQ() # Rational numbers

$\mathbb{Z}_n$, integers mod $n$:





In [7]:



    


Z_2Ring = Integers(2)
Z_16Ring = Integers(16)

Quadratic Field is the the field formed by combining the rationals with a solution to polynomial equation $x^2-n=0$, $\equiv \mathbb{Q}[\sqrt{n}]$





In [8]:



    


QuadField2Ring = QuadraticField(2)



    











In [9]:



    


[method for method in dir(ZZRing) if method[0] is not '_']



    


















    
Out[9]:








['N',
 'abs',
 'additive_order',
 'base_extend',
 'base_ring',
 'binary',
 'binomial',
 'bits',
 'cartesian_product',
 'category',
 'ceil',
 'class_number',
 'conjugate',
 'coprime_integers',
 'crt',
 'db',
 'degree',
 'denominator',
 'digits',
 'dist',
 'divide_knowing_divisible_by',
 'divides',
 'divisors',
 'dump',
 'dumps',
 'euclidean_degree',
 'exact_log',
 'exp',
 'factor',
 'factorial',
 'floor',
 'gamma',
 'gcd',
 'global_height',
 'imag',
 'inverse_mod',
 'inverse_of_unit',
 'is_idempotent',
 'is_integer',
 'is_integral',
 'is_irreducible',
 'is_nilpotent',
 'is_norm',
 'is_one',
 'is_perfect_power',
 'is_power_of',
 'is_prime',
 'is_prime_power',
 'is_pseudoprime',
 'is_pseudoprime_power',
 'is_square',
 'is_squarefree',
 'is_unit',
 'is_zero',
 'isqrt',
 'jacobi',
 'kronecker',
 'lcm',
 'leading_coefficient',
 'list',
 'log',
 'mod',
 'multifactorial',
 'multiplicative_order',
 'n',
 'nbits',
 'ndigits',
 'next_prime',
 'next_prime_power',
 'next_probable_prime',
 'nth_root',
 'numerator',
 'numerical_approx',
 'odd_part',
 'ord',
 'order',
 'ordinal_str',
 'parent',
 'perfect_power',
 'popcount',
 'powermod',
 'powermodm_ui',
 'powers',
 'previous_prime',
 'previous_prime_power',
 'prime_divisors',
 'prime_factors',
 'prime_to_m_part',
 'quo_rem',
 'radical',
 'rank',
 'rational_reconstruction',
 'real',
 'rename',
 'reset_name',
 'save',
 'sign',
 'sqrt',
 'sqrtrem',
 'squarefree_part',
 'str',
 'subs',
 'substitute',
 'support',
 'test_bit',
 'trailing_zero_bits',
 'trial_division',
 'val_unit',
 'valuation',
 'version',
 'xgcd']

















In [10]:



    


ZZRing.rank()



    


















    
Out[10]:








0

ring of Gaussian integers, $\mathbb{I}_m \equiv \mathbb{Z}/m\mathbb{Z}$





In [1]:



    


ZZ[I]



    


















    
Out[1]:








Order in Number Field in I with defining polynomial x^2 + 1

















In [3]:



    


ZZ[I].is_integral_domain()



    


















    
Out[3]:








True

















In [4]:



    


ZZ[I].zero()



    


















    
Out[4]:








0

















In [5]:



    


ZZ[I].one()



    


















    
Out[5]:








1

















In [11]:



    


ZZ[I].is_ring()



    


















    
Out[11]:








True

Polynomials

For reading - cf. Sec. 3.4 Greatest Common Divisors, of Joseph Rotman's Advanced Modern Algebra

For the Sage Math exercises - cf. in Abstract Algebra - Theory and Applications by Thomas W. Judson, 17 Polynomials, 17.6 Sage

\begin{gathered} \mathbb{I}_8 \equiv \mathbb{Z}/8\mathbb{Z} \text{ is the (commutative) ring } \\ \mathbb{Z}/8\mathbb{Z}[x] \text{ is the polynomial ring over } \mathbb{Z}/8\mathbb{Z} \end{gathered} 




In [12]:



    


R.<x> = Integers(8)[]; R



    


















    
Out[12]:








Univariate Polynomial Ring in x over Ring of integers modulo 8

















In [19]:



    


R1.<x> = PolynomialRing(ZZ.quo(8*ZZ)); R1



    


















    
Out[19]:








Univariate Polynomial Ring in x over Ring of integers modulo 8

















In [20]:



    


R1 == R



    


















    
Out[20]:








True

















In [21]:



    


S.<y> = ZZ[]; S



    


















    
Out[21]:








Univariate Polynomial Ring in y over Integer Ring

















In [22]:



    


T.<z> = QQ[]; T



    


















    
Out[22]:








Univariate Polynomial Ring in z over Rational Field

















In [23]:



    


R.is_finite()



    


















    
Out[23]:








False

















In [25]:



    


R.is_integral_domain()



    


















    
Out[25]:








False

















In [26]:



    


S.is_integral_domain()



    


















    
Out[26]:








True

















In [27]:



    


T.is_field()



    


















    
Out[27]:








False

















In [28]:



    


R.characteristic()



    


















    
Out[28]:








8

















In [29]:



    


T.characteristic()



    


















    
Out[29]:








0

















In [31]:



    


print(y in S)
x in S



    


















    






True










    
Out[31]:








False

















In [32]:



    


q = (3/2) + (5/4)*z^2 
q in T



    


















    
Out[32]:








True

















In [34]:



    


print(3 in S)
r = 3
print( r.parent() )
s = 3*y^0
s.parent()



    


















    






True
Integer Ring










    
Out[34]:








Univariate Polynomial Ring in y over Integer Ring

Polynomials can be evaluated like they are functions





In [35]:



    


p = 3 + 5*x + 2*x^2
print( p.parent() )
p(1)



    


















    






Univariate Polynomial Ring in x over Ring of integers modulo 8










    
Out[35]:








2

















In [36]:



    


[p(t) for t in Integers(8)]



    


















    
Out[36]:








[3, 2, 5, 4, 7, 6, 1, 0]

















In [38]:



    


g = 4*x^2 + 4*x
[g(t) for t in Integers(8)]



    


















    
Out[38]:








[0, 0, 0, 0, 0, 0, 0, 0]

















In [39]:



    


M.<s,t> = QQ[]; M



    


















    
Out[39]:








Multivariate Polynomial Ring in s, t over Rational Field

Irreducible Polynomials





In [40]:



    


R.<x> = QQ[]
p = 1/4*x^4 - x^3 + x^2 - x - 1/2
p.is_irreducible()



    


















    
Out[40]:








True

















In [41]:



    


p.factor()



    


















    
Out[41]:








(1/4) * (x^4 - 4*x^3 + 4*x^2 - 4*x - 2)

















In [42]:



    


q = 2*x^5 + 5/2*x^4 + 3/4*x^3 - 25/24*x^2 - x - 1/2
q.is_irreducible()



    


















    
Out[42]:








False

















In [43]:



    


q.factor()



    


















    
Out[43]:








(2) * (x^2 + 3/2*x + 3/4) * (x^3 - 1/4*x^2 - 1/3)

















In [51]:



    


F.<a> = FiniteField(5^2)
S.<y> = F[]
p = 2*y^5 + 2*y^4 + 4*y^3 + 2*y^2 + 3*y + 1
p.is_irreducible()



    


















    
Out[51]:








True

















In [52]:



    


p.factor()



    


















    
Out[52]:








(2) * (y^5 + y^4 + 2*y^3 + y^2 + 4*y + 3)

















In [54]:



    


q = 3*y^4+2*y^3-y+4; print( q.is_irreducible() )
q.factor()



    


















    






False










    
Out[54]:








(3) * (y^2 + (a + 4)*y + 2*a + 3) * (y^2 + 4*a*y + 3*a)

















In [55]:



    


r = y^4 + 2*y^3 + 3*y^2+4; r.factor()



    


















    
Out[55]:








(y + 4) * (y^3 + 3*y^2 + y + 1)

















In [57]:



    


s = 3*y^4+2*y^3-y+3; s.factor()



    


















    
Out[57]:








(3) * (y + 1) * (y + 3) * (y + 2*a + 4) * (y + 3*a + 1)

















In [58]:



    


F.modulus()



    


















    
Out[58]:








x^2 + 4*x + 2

















In [60]:



    


F.modulus().is_irreducible()



    


















    
Out[60]:








True

Polynomials over Fields

If $F$ is a field, then every ideal of $F[x]$ is principal. cf. (http://abstract.ups.edu/aata/poly-sage.html)





In [61]:



    


W.<w> = QQ[]
r = -w^5 + 5*w^4 - 4*w^3 + 14*w^2 - 67*w + 17
s = 3*w^5 - 14*w^4 + 12*w^3 - 6*w^2 + w
S = W.ideal(r, s)
S



    


















    
Out[61]:








Principal ideal (w^2 - 4*w + 1) of Univariate Polynomial Ring in w over Rational Field

















In [62]:



    


(w^2)*r + (3*w-6)*s in S



    


















    
Out[62]:








True

















In [63]:



    


F = Integers(7)
R.<x> = F[]
p = x^5 + x + 4
p.is_irreducible()



    


















    
Out[63]:








True

cf. Sage Reference Manual: Polynomials, Release 7.3





In [64]:



    


_.<x> = PolynomialRing(ZZ)
f = x^4 + 2*x^2 + 1
print( f.coefficients() )
f.coefficients(sparse=False)



    


















    






[1, 2, 1]










    
Out[64]:








[1, 0, 2, 0, 1]

















In [65]:



    


f.degree()



    


















    
Out[65]:








4

Take derivatives of polynomials

$\begin{gathered} f'(x) \\ f''(x) \end{gathered}$





In [66]:



    


print( f.derivative(x) )
print( f.derivative(x,x) )
f.derivative(x,2)



    


















    






4*x^3 + 4*x
12*x^2 + 4










    
Out[66]:








12*x^2 + 4

cf. J. Rotman, Advanced Modern Algebra, Exercise 3.28





In [89]:



    


ZZ5_x_polys.<x> = PolynomialRing( Integers(5) )



    











In [92]:



    


( Integers(5)(1)*x**3-Integers(5)(7)*x+Integers(5)(6) ).gcd( Integers(5)(1)*x**2-Integers(5)(1)*x-Integers(5)(2) )
(x^3-7*x+6).gcd(x^2-x-2)



    


















    
Out[92]:








x + 3

















In [95]:



    


print( (x^2-x-2).factor() )
(x^3-7*x+6).factor()
factor( x^3-7*x+6)



    


















    






(x + 1) * (x + 3)










    
Out[95]:








(x + 4) * (x + 3)^2

















In [93]:



    


print( (x+1)*(x+3) )



    


















    






x^2 + 4*x + 3

















In [96]:



    


p0328 = x^2 - x - 2
q0328 = x^3 - 7*x+6
print( p0328.parent() )
q0328.parent()



    


















    






Univariate Polynomial Ring in x over Ring of integers modulo 5










    
Out[96]:








Univariate Polynomial Ring in x over Ring of integers modulo 5

At this point, this helped to check what's going on: cf. Ring Z/nZ of integers modulo n





In [94]:



    


print( Integers(5).0 )
print( Integers(5)(0) + Integers(5)(3) )
print( Integers(5)(0) + Integers(5)(6) )
print( Integers(5)(0)*Integers(5)(2) )
print( Integers(5)(1)*Integers(5)(3) ) 

print( 0 )
print( 0 + 3 )
print( 0 + 6 )
print( 0 * 2 )
print( 1 * 3 )



    


















    






1
3
1
0
3
0
3
6
0
3

cf. J. Rotman, Advanced Modern Algebra, Exercise 3.30





In [97]:



    


ZZ3_x_polys.<x> = PolynomialRing( Integers(3) )



    











In [98]:



    


f0330 = x^2 + 1
g0330 = x^3 + x + 1



    











In [99]:



    


f0330.gcd( g0330 )



    


















    
Out[99]:








1

















In [101]:



    


print( factor( f0330 ) )
factor( g0330 )



    


















    






x^2 + 1










    
Out[101]:








(x + 2) * (x^2 + x + 2)

Modules





In [ ]:



    






    











In [ ]:



    






    











In [11]:



    


%display latex



    











In [12]:



    


RingModule = Modules(Rings())



    











In [13]:



    


[method for method in dir(RingModule) if method[0] is not '_']



    


















    
Out[13]:























In [14]:



    


RingModule.category_graph()



    


















    
Out[14]:
























In [15]:



    


RingModule.axioms()



    


















    
Out[15]:























In [16]:



    


ZZmodule = Modules(ZZ)



    











In [17]:



    


ZZmodule.AdditiveInverse(), ZZmodule.AdditiveUnital(), ZZmodule.an_instance()



    


















    
Out[17]:























In [18]:



    


ZZmodule.category_graph()



    


















    
Out[18]:

cf. Chapter 7 Modules and Categories, 7.1 Modules of Rotman, Advanced Modern Algebra, pp. 423, Example 7.1





In [19]:



    


n = var('n',domain="integer")
RRmoduleVecSpace = VectorSpace(RR,3)



    











In [20]:



    


[method for method in dir(RRmoduleVecSpace) if method[0] is not '_']



    


















    
Out[20]:























In [21]:



    


RRmoduleVecSpace.basis(), RRmoduleVecSpace.basis_matrix(), RRmoduleVecSpace.gens()



    


















    
Out[21]:























In [22]:



    


RRmoduleVecSpace.gens_dict()



    


















    
Out[22]:























In [ ]:

Content source: ernestyalumni/qSApoly

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