Beginning in the fall of 1957.

which element is necessary to complete the sentence?

independent clause
subject
subordinate clause
predicate

The given statement is an example of a phrase, specifically, a participial phrase. The element that is necessary to complete the sentence is an independent clause. A participial phrase consists of words with the -ing or -en form plus its modifiers. This kind of phrase always functions as an adjective. The phrase "Beginning in the fall of 1957" is describing a certain noun, which is found in the independent clause.

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