It's very hard to prove a counterfactual like 'could the English have conquered France'. We know they didn't, but whether they could have succeeded requires a lot of speculation. In theory, had the people and nobles of France rejected the Valois monarchy and embraced the Kings of England then sure, maybe Edward III or Henry V could have become king, but without that sudden turn in French opinion the odds were always against the English.

One of points that noted military historian Michael Prestwich makes in his recent history of the Hundred Years War is that what should be remarkable is not that France drove England out, but that England succeeded in conquering as much as it did. England is a much smaller country than France, and during the Middle Ages its population was far smaller than France. England certainly had several advantages, its political structure being more solid and developed than France’s is a big one but being smaller always made conquering France a problem for England. Winning battles is an important aspect of war, but it doesn’t really matter how much territory you conquer if you can’t supply the soldiers to hold it, and holding territory was a huge challenge for England. England had neither the soldiers nor (as the war progressed) the money to garrison all the castles they took and reliably field armies in all of the regions where France could possibly launch a counteroffensive. Occupations and nation building have always been hard, even in the Middle Ages.

Let’s talk some specifics, though. I’m going to focus on the period of the war beginning with Henry V, since that’s generally seen as the period when England came the closest to conquering France. One thing that’s worth emphasising right out of the gate is that Henry V’s conquests were not solely the result of him being an amazing general. He was certainly a capable commander (although I’ve been to at least one paper that used his failures in the Scottish Wars to point out limitations of his ability), but at the time of his greatest success he was assisted by the fact that France was in a full-blown civil war.

The short version is that while Charles V, King of France, had succeeded in taking back substantial power and lands from the Kings of England in the wake of the Treaty of Bretigny in 1360, his son Charles VI’s career was cut short by the onset of ‘madness’. What mental illness Charles had is impossible to know, and diagnosing specific mental illnesses in historical figures is very problematic, but he certainly seems to have suffered from severe bouts of mental delusions and illness that rendered him incapable of ruling for extended periods of time and fundamentally broke his power as a ruler in general. In the wake of his madness two factions rose in French politics to compete for who would rule the kingdom. One, the House of Orleans more commonly called the Armagnacs for complicated reasons, was led by the Duke of Orleans who was Charles VI’s younger brother. The other, the Burgundians, was led by John the Fearless, Duke of Burgundy, who was Charles’ cousin (John’s father was Philip the Bold who was Charles V’s brother. Technically Philip was in charge of the Burgundians at the start of the conflict, but he died in 1404 and much of the Civil War happened during John’s life).

Initially the two factions did the usual medieval politics thing of vying for control of the King and generally causing trouble for each other with only limited bloodshed. Things really fell apart in 1407 when John the Fearless had Louis I, Duke of Orleans, assassinated. A few years later the Burgundians captured Paris, and with that they took control of Charles VI. This led to a major breakdown between the factions (as you might expect) and left a great opportunity for England to get involved. The two factions actually managed to put their differences aside to oppose Henry V’s invasion in 1415, which culminated in France’s disastrous defeat at Agincourt. At the battle Louis’ son Charles (now Duke of Orleans) was captured by the English and would spend much of the war an English captive, something that hurt his faction and benefited the Burgundians. Even with all this conflict the two French parties were resistant to fully allying with England (despite the English being very interested in doing so) up until 1419. In 1419, as an act of revenge for the death of Louis, John the Fearless was assassinated by Armagnac forces.

In response to his father’s death, Philip the Good (now Duke of Burgundy) signed the Treaty of Troyes in 1420, officially allying the Burgundians with the English. The treaty also famously declared that Henry would inherit the monarchy upon the death of Charles VI and disinherited the (future) king Charles VII of his rights.

It is during this period that Henry V conquered vast swaths of France with substantial Burgundian support. This support didn’t always take the form of soldiers (although Burgundians were often found in English armies at this time), it also included the more basic fact that Burgundian territory protected areas of English control and allowed for England to focus its limited resources onto certain areas without concern.

Henry V’s death in 1422 was certainly a significant blow to the English, but it is worth remembering that English generals actually achieved several significant victories in the years that followed, most notably at the Battle of Verneuil in 1424, and English held territory actually expanded in the years following Henry’s death. However, the Burgundian/English alliance wasn’t always on the firmest footing, and the new infant King Henry VI was obviously not the same forceful political personality his father had been. The appearance of Joan of Arc in 1429 and the push back of English control (which was in part the result of Joan and her commander’s strategy, and in part due to England being unable to put significant garrisons in all their castles to oppose Joan’s frontal assaults) started to put some cracks in the Anglo-Burgundian alliance. The alliance still outlived Joan, though, as she was captured by Burgundians and sold to the English to be burned for heresy, but only a few years after her death, in 1435 King Charles VII signed a peace treaty with Philip the Good ending the Anglo-Burgundian alliance.

With the alliance over the writing was on the wall for England, really. Not only did they not have Burgundy to shore up their defences, they now had to deal with attacks from their former allies. Charles VII also managed to restructure and reorganise his finances and military to better wage his war at the same time that England was slowly descending into a civil war of its own. While it took nearly 20 years for France to retake all of its territory, there was little England could do to oppose the steady march of French military resources. England’s finances had been pushed to their limits to get them as far as they had, and without French lands to subsidize the war there was little stomach for paying for it in England. Convincing the people of England to finance a foreign war for their king was always a struggle, especially when they were losing. Victory in battle brought money and prestige, slowly losing a series of brutal sieges brought neither. Obviously English interests in France didn’t end in 1453, Henry VIII led an invasion less than a century later, but the outbreak of the War of the Roses certainly put an end to English interests in France for a while and they never again came anywhere near as close to conquering France as they did. A unified France just presented too great a challenge for England to realistically conquer without significant support from within the Kingdom, something they never got again after the end of the Anglo-Burgundian treaty.

David Green’s The Hundred Years War: A People’s History is probably the best overall introduction to the conflict and its many facets.

Michael Prestwich’s A Short History of the Hundred Years War is also great, and much more of a military history if that’s your vibe.

Anne Curry’s The Hundred Years War is a classic history, and her work on medieval soldiers and the Battle of Agincourt cannot be understated.

Kelly DeVries’ Joan of Arc: A Military Leader is a great military history of the most famous female general in history and also includes a lot of good detail about the Burgundian alliance.