We must remember that two people with the same education and knowledge of original Biblical languages, will, and commonly do, conclude opposite conclusions while maintaining proclaimed objectivity in their exegesis. This is exactly what Protestant and Catholic scholars do concerning this verse. The reality is that everyone makes their exegetical conclusion conform to their hermeneutic, whether they admit it, or not. So maybe, in the end this question is best answered ultimately at C.SE but at the same time some interesting aspects will be brought up here that you would probably not find there.
Anyway to try and help outline a bit of the exegetical aspect of the verse, without claiming any magical objectivity let me just state the obvious. A literal word for word translation of the Greek would not include the word ‘alone’. The Greek simply says:
λογιζομεθα (We reckon) ουν (therefore) πιστει (by faith) δικαιουσθαι (to be justified) ανθρωπον (a man) χωρις (apart from) εργων (works of) νομου (law) (Newberry, T., & Berry, G. R. (2004). The interlinear literal translation of the Greek New Testament, Ro 3:28)
I looked up the Peshitta and it has the same simple format as the Greek without a word for 'alone' added.
However most translations are not literal word-for-word renderings as much of the ideas are lost that way. Rather most translations express original ideas into different words of another language. From a linguistic perspective, where the idea is being translated, rather than each literal word, Luther's rendering is certainly a valid possibility. The idea contained in the phrase 'achievement by means of X and not by means of Y' is arguably the equivalent of 'achievement by means of X alone and not Y'. They are communicating the same concept, one just highlights the exclusivity of X which should be already understood from a rule of logic in the former rendering.
For example, Charles Hodge a former professor of Biblical Languages at Princeton says Luther’s translation is obviously very accurate from the standpoint of a simple 'idea translation' in fact other's before Luther made the translation in the same way:
From the nature of the case, if justification is by faith, it must be by faith alone. Luther’s version, therefore, allein durch den glauben, is fully justified by the context. The Romanists, indeed, made a great outcry against that version as a gross perversion of Scripture, although Catholic translators before the time of Luther had given the same translation. So in the Nuremberg Bible, 1483, “Nur durch den glauben.’ And the Italian Bibles of Geneva, 1476, and of Venice, 1538 per sola fede. The Fathers also often use the expression, “man is justified by faith alone;” so that Erasmus, De Ratione Concionandi, Lib. III., says, “Vox sola, tot clamoribus lapi data hoc sæculo in Luthero, reverenter in Patribus auditur.” See Koppe and Tholuck on this verse. (Hodge, C. (1882). A commentary on the Epistle to the Romans, p. 156)
So what was Luther up to? Luther was probably choosing his words carefully to translate what he thought the true meaning of the verse was, knowing full well that opposing theories would be more offended and more manifest by the crucial distinction of the word 'alone'. If he was right, or wrong in doing so would take much more analysis then this one verse, or any one single verse. This really leads us back then to the subject of the reformation and back to a site like C.SE for more debate over the real doctrinal issues at stake.
As you can probably tell, I am approaching the question from a Protestant hermeneutical tendency.