Ferrara  listen (help·info) is a city and comune in Emilia-Romagna, northern Italy, capital city of the Province of Ferrara. It is situated 50 km north-northeast of Bologna, on the Po di Volano, a branch channel of the main stream of the Po River, located 5 km north. The town has broad streets and numerous palaces dating from the 14th century and 15th century, when it hosted the court of the House of Este. For its beauty and cultural importance it has been qualified by UNESCO as World Heritage Site. Modern times have brought a renewal of industrial activity. Ferrara is on the main rail line from Bologna to Padua and Venice, and has branches to Ravenna, Poggio Rusco (for Suzzara) and Codigoro. In 2006, due to its important historical significance, Ferrara became the headquarters of the Italian Hermitage Museum. It is the fifth city in the world to have been linked with the Russian museum. From this union was born the Hermitage-Italy Foundation.

The origin of Ferrara is uncertain, it was probably settled by the inhabitants of the lagoons at the mouth of Po river; there are two early centers of settlement, one round the cathedral, the other, the castrum bizantino, being the San Pietro district, on the opposite shore, where the Primaro empties into the Volano channel. Ferrara appears first in a document of the Lombard king Desiderius of 753 AD, as a city forming part of the Exarchate of Ravenna. Desiderius pledged a Lombard ducatus ferrariae ("Duchy of Ferrara") in 757 to Pope Stephen II. After 984 it was a fief of Tedaldo, count of Modena and Canossa, nephew of the emperor Otto I. It afterwards made itself independent, and in 1101 was taken by siege by the countess Matilda. At this time it was mainly dominated by several great families, among them the prominent Adelardi (or Aleardi) family.

In mathematics the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, p-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p - the higher the power the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

p-adic numbers were first described by Kurt Hensel in 1897,, though with hindsight some of Kummer's earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.