Translation:She is the professor I wrote to.
"She is the professor whom I have written" has not been accepted. Instead Duo suggested "She is the professor I wrote to." Now you can write to me, but you can also just write me, at least in American English:
https://www.oxfordlearnersdictionaries.com/definition/english/write?q=write (look for "write somebody" under the third meaning).
This needs to be added.
Why is it not"die ich geschrieben habe", because it's referred to the professor (who's a female)
In German you can write a book, a letter, a sentence, but not a person. It would have to be "to the professor", which is expressed with the dativ case, instead of the accusative, which is only for direct objects like a letter etc.
See also http://german.speak7.com/german_articles.htm - "der" is not always masculine, it appears in multiple positions in the table.
I had it marked wrong but I am not sure why. With the latest update it seems I can no longer check what I had written.
So what would you say instead? She is the professor to who I wrote?
And what about this https://blog.oxforddictionaries.com/2011/11/28/grammar-myths-prepositions/ ? I use sentences like those more or less often, so it would be interesting to know if they sound odd.
Better English is simply, "She is the professor that I wrote." Many people end sentences with prepositions, especially when speaking, but it is considered bad form. (I'm an English teacher.)
Yes, what you learned is correct. However, if you leave out the "that" and switch it for "who/whom" it ends up sounding stilted and very, very formal. ("She is the professor (to) whom I have written") It is grammatically correct, but would really only be spoken by a non-native speaker or someone who is purposefully trying to sound very formal or even snooty. You could say, "She is the professor who I wrote" which sounds more colloquial, but it is incorrect because this sentence requires "whom". I think many people use "that" because it avoids the whole "who/whom" issue.
At least to my ear, "She is the professor (to) whom I wrote" sounds infinitely better than "She is the professor that I wrote." Particularly since you can't really attach "to" to "that" here, it sounds too much like "This is the paper that I wrote" - clearly not the intended meaning. So avoiding "that" with people is still very much a good idea.
“This is the sort of bloody nonsense up with which I will not put.”
- attributed to Winston Churchill.
More seriously, educate yourself before declaring something bad English: