Does Islam allow the use of child soldiers?
As an example, the use of five year olds in Syria.
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Well a few months ago I read this hadith which shows that the right age where our Messenger (peace be upon him) accepted young boys to join in to fight in Jihad is 15 years, which is also considered as the minimum age according to this Fatwa! but in some cases the Messenger of Allah (peace be upon him) accepted some younger boys like Samara ibn Jundub (May Alalh be pleased with him) who was very strong and Rafi' ibn Khodayj (May peace be upon him) for being at shooting arrows good whom both participated at Uhud!
But in any case a younger boy shouldn't fight unless he has his parents agreement. As one can conclude from the following hadith about a young man who wanted to fight or in general from a hadith narrated by many sahab (May Allah be pleased with them):
I asked Allah's Messenger (ﷺ), "O Allah's Messenger (ﷺ)! What is the best deed?" He replied, "To offer the prayers at their early stated fixed times." I asked, "What is next in goodness?" He replied, "To be good and dutiful to your parents." I further asked, what is next in goodness?" He replied, "To participate in Jihad in Allah's Cause." I did not ask Allah's Messenger (ﷺ) anymore and if I had asked him more, he would have told me more. see for example here or here.
This is all I could tell to answer your question but be aware that there are many other rules to fix whether one could join a fighting Army or not (for example what if your family will hunger? if you have debts? is it allowed to fight without permission or call of an Authority?)