The mathematical insights made by Paul Dirac were essential to his productive collaboration with Edwin Schrodinger, with whom they shared a Nobel prize in 1933, and to his own successful efforts to apply the special theory of relativity to the quantum mechanics of the electron.

A to his productive collaboration with Edwin Schrodinger, with whom they shared a Nobel prize in 1933, and to
B to his productive collaboration with Edwin Schrodinger, with whom Dirac shared a Nobel prize in 1933, and to
C to his productive collaboration with Edwin Schrodinger, with whom both men shared a Nobel prize in 1933, and to
D towards his productive collaboration with Edwin Schrodinger, with whom Dirac shared a Nobel prize in 1933, and to
E in his productive collaboration with Edwin Schrodinger, with whom Dirac shares a Nobel prize in 1933, and in
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Hi Daagh,

If in E expression is - Dirac shared in 1993 - then which option will be preferred B or E..

The mathematical insights made by Paul Dirac were essential to his productive collaboration with Edwin Schrodinger, with whom they shared a Nobel prize in 1933, and to his own successful efforts to apply the special theory of relativity to the quantum mechanics of the electron.
A to his productive collaboration with Edwin Schrodinger, with whom they shared a Nobel prize in 1933, and to
Use of THEY is incorrect since such use means that there was someone else, apart from Paul and Edwin, who shared a Nobel Prize, an idea contrary to original intent that Paul shared Nobel prize with Edwin.
B to his productive collaboration with Edwin Schrodinger, with whom Dirac shared a Nobel prize in 1933, and to
Correct
C to his productive collaboration with Edwin Schrodinger, with whom both men shared a Nobel prize in 1933, and to
Use of BOTH is incorrect on the lines as explained in option A.
D towards his productive collaboration with Edwin Schrodinger, with whom Dirac shared a Nobel prize in 1933, and to
ESSENTIAL TO is correct use.
E in his productive collaboration with Edwin Schrodinger, with whom Dirac shares a Nobel prize in 1933, and in
ESSENTIAL TO is correct use.
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"Essential TO" is the correct Idiom - "essential IN" is wrong.

You have to memorise the idioms, there are no rules for them.