A dissenting view from the other answers: **Yes, it is fine to silently correct obvious typos that do not affect the subject matter.**  (But only when you are 100% confident that they really are just trivial typos.)

In non-academic contexts, this is pretty much universal practice. The Chicago Manual of Style, for instance, says “Obvious typographic errors may be corrected silently (without comment or sic) unless the passage quoted is from an older work or a manuscript source where idiosyncrasies of spelling are generally preserved.” and I do not know any major style guide that differs from this.

In an academic setting, **you should certainly be extremely cautious in judging what’s really a typo**, as comments on the question point out.  However, you should usually be well-qualified to judge this, as an academic in a field closely related to that of the writers you’re quoting.

So I see no positive reason to treat the academic case differently from the non-academic.  Scientific accuracy and clarity is paramount; literal typographical fidelity is no more important in academia than in most other fields.

Meanwhile, all the negatives of replicating the typo still apply.  Leaving it in without a “[sic]” is distracting to the reader, and also makes it unclear whether the typo is due to you or the original authors.  Adding a “[sic]” is even more distracting to the reader, is a bit harsh towards the original authors (drawing attention to a trivial mistake they made), and may be read as intentionally disrespectful to them.

**Edit:** What is “an obvious typo”?  General-purpose style guides give guidelines like “something which you are absolutely confident the author would have corrected, had they noticed it; and which can’t be read in any other way than the ‘corrected’ way”.  This principle still seems completely appropriate in academic settings, with the caveats that academic writing is particularly likely to include unusual terminology or deliberately-chosen subtleties of wording, and (again) of erring on the side of caution, since accuracy is critical.

Original: `Andrew Wiles’ famous profo of Fermat’s last theorem…`

Suggestion: check that `profo` is not a technical or facetious term that the author is using elsewhere in the text, or that appears in other literature.  Having ascertained that, correct `profo` -> `proof`.

Original: `Andrerw Wiles’ famous proof of Fermat’s last theorem…`

Suggestion: check that there is not some mathematician called “Andrerw Wiles” who gave another proof of FLT. Having ascertained that, correct `Andrerw` –> `Andrew`.

Original: `Andrew Wiles’ famous proof of Fermats last theorem…`

Suggestion: check that the author is not deliberately using non-standard punctuation elswhere in the text.  Having ascertained that, correct `Fermats` -> `Fermat’s`.

Original: `Andrew Wiles’ famous proof of Fermat’s little theorem…`

Suggestion: Do not correct.  This is almost certainly a typo — “Fermat’s little theorem” does exist, but Wiles’ famous proof is of the last not the little — but it affects the subject matter non-trivially, in that the incorrect reading still makes sense.

Original: `Andrew Wiles’ proof famous of Fermat’s last theorem…`

Suggestion: Do not correct.  This is most likely either an editing typo or a simple non-native speaker mistake; but it is conceivable that an author chose this wording deliberately.