What purpose does the subplot of the relationship between Polonius and his children serve?

The subplot of Polonius and his children highlights the lack of genuine family relationships at the Danish court. The subplot also serves as a dramatic foil for Hamlet's behavior, proving that not all members of Elsinore's royal court are mad or evil.

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The subplot of Polonius' relationship with his children highlights the lack of genuine family relationships at the Danish court. As with everything else at Elsinore, power, not love, is what determines relations between Polonius and his children. We can never know for sure if Polonius does indeed love his children. If so, he has a funny way of showing it.

Once he's bid farewell to Laertes, who's off to Paris for his studies, Polonius immediately instructs one of his lackeys to go with his son to France, where he is to spy on him. Once there, he's to put it about that Laertes is involved in a debauched lifestyle. How people react to such news will determine whether or not Laertes really has been a naughty boy. Most normal parents would balk at such ruthless manipulation of their children's lives. But not Polonius. As a hardened veteran of the Danish court, such power moves are second nature to him.

Even worse is his manipulation of Ophelia, whom he shamefully uses as a pawn in an elaborate game to determine the reasons behind Hamlet's unusual behavior. Ophelia is emotionally destroyed by the experience, yet Polonius doesn't seem to care. As always, he's only interested in the political consequences of Hamlet's vicious rant against his daughter.

Once again, we see how proximity to power distorts family relationships, which, as with all royal courts, are determined by power rather than by bonds of love and affection.

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The subplot involving Polonius, Ophelia, and Laertes is crucial to the central plot in several ways.

First, the relationship between Laertes and Polonius parallels that between Hamlet and his dead father; in this respect, Laertes serves as a foil for Hamlet. The theme that a son must avenge his father’s death joins the two stories. Their duel kills both young men.

Second, Polonius’s position at court and his hypocritical behavior emphasize the play’s overall theme of corruption: he exemplifies what is “rotten” in the Danish court. Incapable of following his own advice, “to thine own self be true,” Polonius is a total sycophant. His complicity in the monarch’s schemes, by hiding behind the arras, leads to his death.

Third, Polonius’s treatment of his daughter and his acquiescence of the monarchy using her to bait and spy on Hamlet, while generally related to the hypocrisy theme, is crucial to the main plot. Her father’s behavior endangers Ophelia by making Hamlet further suspect Claudius and Gertrude’s plot and also helps destroy her relationship with Hamlet. This directly or indirectly leads to her death. Depending on one’s interpretation, as Shakespeare leaves it ambiguous, either Gertrude kills Ophelia by pushing her the river, or Ophelia kills herself by jumping in.

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One of the interesting ideas explored in the play is family relationships.  Hamlet and his relationship with his family is broken.  The relationship of Polonius and his children, on the other hand, seems like a good one.

Does he love his children?  Yes. ...

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Does he know how to love them?  Maybe not.

On the surface, Polonius would appear to be a doddering old fool but the advice he gives is very good advice.  He is like any father sending his young son away to school.  Does he trust his son?  Of course not since most men remember their own youthful indiscretions  But most fathers don't send out a spy.

His relationship with his daughter is different.  Like any father, no young man is good enough and that includes Hamlet.  How long he has been a single parent isn't known but being a single father to a young female could not be easy.

A more telling bit of information is how he addresses his children.  Does he use the familiar thee or thou or the more generalized you?

As for the importance of the subplot, without these three characters, the play would be missing a love interest for Hamlet, a foil for Hamlet and a father who wants only the best for his children as opposed to a step father who wants Hamlet dead.

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Polonius' relationship with Laertes and Ophelia is a subplot which heightens the tones of dishonesty and distrust in the play, and speaks directly to a theme of corrupted authority.

First of all, after a lengthy speech given to Laertes about morals and virtues, Polonius then sends a servant to go spy on his son while in school.  He even plans out the lie this servant should tell in order to fool Laertes.  Later, Polonius hides behind a curtain to spy on his love-sick daughter and Hamlet.

While it is a little difficult to take Polonius' character seriously, the hypocrisy and humor that he provides to the story is no doubt intentional.  There is dramatic irony in the fact that Shakespeare builds him up to be disliked because of (among other things) the way he treats his children, and then has his son fighting at the end to avenge his death.  Polonius' lack of trust of his children is likely driven by his own propensity to lie.  He is a minor character who contributes to the bigger story in a subtle but very specific way.

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