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""" 

TESTS:: 

 

sage: R = Zp(5, prec=5, type='fixed-mod') 

sage: a = random_matrix(R,5) 

sage: a.determinant() 

1 + 3*5 + 3*5^2 + 2*5^4 + O(5^5) 

sage: K = Qp(3, 10,'capped-rel'); K.krull_dimension() 

0 

 

Computation of logs:: 

 

sage: Qp(5)(1).log() 

O(5^20) 

sage: Qp(5)(-1).log() 

O(5^20) 

sage: Qp(5,prec=5)(7).log() 

4*5^2 + 4*5^3 + 3*5^4 + O(5^5) 

sage: Qp(5,prec=10)(25*8).log(p_branch = 0) 

5 + 3*5^3 + 3*5^4 + 2*5^5 + 5^6 + 2*5^7 + 3*5^8 + 3*5^9 + O(5^10) 

sage: Zp(5,prec=10)(8).log() 

5 + 3*5^3 + 3*5^4 + 2*5^5 + 5^6 + 2*5^7 + 3*5^8 + 3*5^9 + O(5^10) 

 

Loading and saving elements of various types:: 

 

sage: a = Zp(5)(-3); loads(dumps(a)) == a 

True 

 

sage: M = MatrixSpace(pAdicField(3,100),2) 

sage: (M([1,0,0,90]) - (1+O(3^100)) * M(1)).left_kernel() 

Vector space of degree 2 and dimension 1 over 3-adic Field with capped relative precision 100 

Basis matrix: 

[1 + O(3^100) 0] 

"""