In Shakespeare's Hamlet, what is the importance of Gertrude being in the play?

In Shakespeare's Hamlet, I believe
that Gertrude is important for several reasons. The most important reason is that
Claudius' marriage to her solidifies Claudius' place as King, when Old Hamlet dies.
Second, she is the connection that brings Hamlet to stay at the
castle. If not for his mother, he most certainly would have returned to school after his
father's funeral if he had not seen the Ghost. If he had seen the
Ghost, I doubt that Claudius would have tolerated Hamlet's behavior very long. I believe
he does so out of deference to his new wife.

Gertrude hasty
remarriage to Claudius after Old Hamlet's death serves to alienate Hamlet even further.
This alienation sets Hamlet up to die because he has so few trustworthy connections; the
peace he makes with his mother comes to late to be of help to him to stop the plotting
that has already begun in earnest to have Hamlet killed. 

Gertrude's
presence, as well as Ophelia's, serves to remind us of the tenuous place women held in
society at that time, and how powerless they were to change or correct what was taking
place in the male-dominated society in which they lived. It is important to understand
why Gertrude is so unaware of Claudius' actions.

Gertrude
may not have been in love with Claudius at all: it was a quick courtship. However, she
is guaranteed protection and support when her husband dies by marrying Claudius, and
more to the point, she may also be the force that allows Hamlet to stay to address his
father's murder. With her marriage, she guarantees a place for Hamlet at
court.