I'm not real knowledgeable on this part of the world at this time in history, so I looked it up and the Dauphin wasn't even at the battle, so I know the movie is making stuff up here. BUT, would a king simply let the common soldiers basically execute the prince of France? Is it in all cases that royalty and nobility were taken prisoner until they could be ransomed off or could a scenario like that of "The King" have realistically happened?
I suppose you could say that there are two separate questions here, one being whether common soldiers might be permitted to kill high nobility on the battlefield, and the other being whether or not they would be allowed to kill a member of the high nobility after the battle had ended, which is essentially the situation at the end of the movie. The need to answer both questions comes from a number of different sources describing Agincourt as the battle where chivalry died in the mud, suggesting that it was substantially different from all previous battles and that the killing of greater nobles was something unusual at the time.
While it's certainly true that the ideal was to take knights and other nobles for ransom, both out of a sense of a shared "club", if you will, and out of financial incentives, this hides a very pragmatic face of battle. If it was impractical to take someone prisoner, either because they were actively fighting you or because your army was outnumbered, then you simply killed them, whether they were a peasant or a king. There were also some special occasions, such as when fighting traitors or against a bitter enemy, when a king might declare that no prisoners were to be taken. Both the French and English declared this at Crecy, Poitiers and Agincourt and, although many high ranking prisoners were taken during the late stages of all three battles, on the whole the death toll among the nobility could be quite shocking.
As an example at the Battle of Crecy, king John of Bohemia, the duke of Lorraine and the counts and lords of Alencon, Flanders, Blois, Harcourt, Salm, Rosenberg and Saint-Venant were all killed, in addition to perhaps as many as two thousand lesser lords and knights. More interestingly, we know of at least four prisoners who were killed after Crecy during a dispute over who had captured them. The details aren't provided beyond that but the source, Colins de Beaumont, was quite possibly a witness to the battle and certainly seems to have helped identify the French dead afterwards.
At Agincourt specifically, three dukes, five counts and nearly 100 other great lords in addition to somewhere around 3600-4000 other knights and men-at-arms. All but two of the major French commanders were killed, along with almost 60% of France's bailiffs and seneschals (officials in charge of Royal justice and administration in a region). The male line of entire families were wiped out, and the region of Picardy was almost devoid of knights and men-at-arms for a generation. This was seen as a great tragedy, but not as something unusual except in scale. After all, large numbers of the great princes (dukes, counts, kings, etc) had been killed at Crecy and Poitiers, so this was nothing new.
While we can never be certain how many of these great lords were killed by common soldiers, there was no outrage over the fact that many were, in fact, killed by lower class men. It was a normal part of warfare for the period, and any lament was less for their deaths than for the lack of a chance for higher ranked men to have killed them and gained honour for themselves.
Coming back around to the Dauphin and The King, had the Dauphin been at Agincourt (he wasn't) and had he shown himself off to Henry after the French had been defeat, Henry wouldn't have ordered his men, whether common or noble, to kill him. The Dauphin was a far too useful political and financial catch to waste by slaughtering him. Even when Henry V ordered the killing of the prisoners (of whom ~700-900 out of 1400-1600 were killed, with ~700 survivors), he deliberately excepted the highest ranking men, such as Arthur de Richemont, the Duke of Brittany. Not only would killing them deny Henry a rich ransom (it was normal for the king to "buy" the highest ranked nobles from their captors), but by keeping them prisoner he could minimise the participation of their lands in future conflict. Without their lord to recruit troops, it would be harder to draw men from those counties and dukedoms.
Capturing the Dauphin, the heir to the throne, would have been even better. Not only would he have been worth a king's ransom (almost literally), but by having him prisoner Henry would be removing a moderating factor in the civil war between the Armagnacs and Burgundians, and gaining an even greater ability to play each side against the other as he strove to become king of France.
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