We describe a new arithmetic system for the Magma computer algebra system for working with p-adic numbers exactly, in the sense that numbers are represented lazily to infinite p-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical and practical justification for its design and describe some use cases. The intention is that this article will be of benefit to anyone wanting to implement similar functionality in other languages.

中文翻译：

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岩浆中的精确p- adic计算

我们将为Magma计算机代数系统描述一种新的算术系统，该系统可以精确地处理p -adic数，从某种意义上说，数字以无穷大的p -adic精度懒散地表示。这是第一个功能强大的此类实现。这具有增加用户友好性和加快某些计算速度，以及强制产生可证明的结果的好处。我们给出其设计的理论和实践依据，并描述一些用例。目的是使本文对希望以其他语言实现类似功能的任何人有所帮助。