What political change was brought about by the fighting of the Hundred Years' War?

One political change in England that resulted from the Hundred Years War was an increase in the pride of the English lower classes that resulted from the crushing defeat of the French nobility at Agincourt. Agincourt was an important battle in the war; it occurred when England ramped up the fighting again after a break in much of the action. Up till this battle, war in Europe was largely a nobleman's game. At Agincourt, lower-class English archers dealt the horsed knights of the French nobility a humiliating and deadly defeat. This decimated the French nobility, who were more or less caught together in the mud and ignobly slaughtered by a rain of arrows; but the English archers who defeated them brought home an important message: that commoners could make a huge difference on the battlefield, and back home in politics as well.

Three main political changes can be discerned as a direct consequence of the Hundred Years' War: the first related to England; the second to France; and the third to both of them together.
In relation to the first, the war led to considerable political turmoil among the English. Their military failures in France caused considerable bitterness and resentment among many English nobles towards the king, Henry VI. Worse still, Henry was a weak, ineffectual ruler, plagued by mental health issues and chronically unable to deal with the developing factions among the nobility. The outbreak of the Wars of the Roses was the almost inevitable result. One war had led to another.
In France, the country's administration became more streamlined and efficient, especially in relation to the collection of taxes, previously a haphazard affair. Necessity dictated the growing centralization of the French state, as the English had still not given up their territorial claims despite their recent setbacks. There was no treaty at the end of the war; this created an atmosphere of great uncertainty in France and a genuine concern that the English would try their luck once more when they decided that the time was right. The French, then, had no choice but to meet this real and ever-present danger. They did this primarily through the establishment of a standing army—still a rarity in those days—one that would be able to withstand a future English invasion more effectively.
In both England and France a growing sense of national identity began to develop as a direct result of the Hundred Years' War. Successive generations on both sides of the English Channel had known nothing else but war; the intense, often barbarous conflict had hardened national loyalties, feeding into a highly potent patriotism. Since the Norman Conquest of England in the eleventh century, the two cultures had merged to a considerable extent, especially among the nobility. But in the aftermath of this savage conflict, they abruptly diverged, establishing a truly national—as opposed to merely an aristocratic—rivalry between England and France that would endure for centuries in one form or another.