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Homework Statement
Why are prime ideals so important?
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Why are prime ideals crucial in algebraic geometry?
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Why are prime ideals so important?
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Hi Metric_Space 
1) Because they form a generalization for prime numbers, and prime numbers are important.
2) Because they correspond to irreducible curves in geometry. For example, the curve y=x2 is irreducible, and indeed, the ideal (y-x2) is prime in [itex]\mathbb{C}[x,y][/itex]. On the other, the curve xy=0 is not irreducible since it exists out of the pieces x=0 and y=0. And indeed, the ideal (xy) is not prime.
1) Because they form a generalization for prime numbers, and prime numbers are important.
2) Because they correspond to irreducible curves in geometry. For example, the curve y=x2 is irreducible, and indeed, the ideal (y-x2) is prime in [itex]\mathbb{C}[x,y][/itex]. On the other, the curve xy=0 is not irreducible since it exists out of the pieces x=0 and y=0. And indeed, the ideal (xy) is not prime.