die Betrachtung
observation
The careful and attentive looking at or observing of something or someone.
Die Betrachtung des alten Gemäldes dauerte mehrere Minuten.
The observation of the old painting lasted several minutes.

observation / consideration
Loading your saved state.
observation
The careful and attentive looking at or observing of something or someone.
Die Betrachtung des alten Gemäldes dauerte mehrere Minuten.
The observation of the old painting lasted several minutes.

The noun 'die Betrachtung' (observation / consideration) is derived directly from the verb 'betrachten' (to look at / consider). If you know the verb, you can infer the meaning of the noun more easily: 'betrachten' means to look at something closely or to think about it.
Er betrachtet das Kunstwerk mit großem Interesse.
He looks at the artwork with great interest.
'Betrachtung' (observation / consideration) is very often used with the genitive to say exactly what is being looked at or thought over. The structure is: 'die Betrachtung' + genitive.
Die Betrachtung der Sterne fasziniert die Menschen seit jeher.
The observation of the stars has always fascinated people.
The context reveals the meaning: If it is about something concrete that can be seen (e.g. a picture, a landscape), visual observation is meant. If it is about an abstract topic (e.g. a question, a problem), mental analysis is meant.
Die stille Betrachtung der Natur beruhigt mich.
The quiet observation of nature calms me.
The expression 'etwas in Betracht ziehen' (to consider something) is very common and means 'etwas erwägen' (to consider something). The thing being considered is in the accusative.
Wir ziehen einen Umzug in eine andere Stadt in Betracht.
We are considering moving to another city.
The expression 'bei näherer Betrachtung' (on closer inspection) often comes at the beginning of a sentence. Then the verb follows directly, and after that the subject (inversion).
Bei näherer Betrachtung erkannte ich die feinen Details des Musters.
On closer inspection, I recognized the fine details of the pattern.
study / degree program
'das Studium' and 'die Betrachtung' can both mean a close engagement with a topic. 'das Studium' is usually deeper and more systematic: you examine something carefully, often with a lot of material or over a longer period of time. 'die Betrachtung' more often means a mental analysis or a particular way of looking at a topic. It can be shorter and often sounds like part of an explanation or a text. When you are researching or examining something for a long time and thoroughly, 'Studium' fits better. When you are thinking about a topic and explaining it from a particular angle, 'Betrachtung' is more natural.

Bei genauer Betrachtung wirkt der Plan gar nicht so teuer.
On closer consideration, the plan does not seem so expensive after all.
Für ihr Studium der Stadtgeschichte liest Anna viele alte Dokumente.
For her study of the city's history, Anna reads many old documents.