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Chapter 1: Problem 1

Suppose that \(k\) is a field of characteristic \(\neq 2\). Decompose intoirreducible components the closed set \(X \subset \mathbb{A}^{3}\) defined by\(x^{2}+y^{2}+z^{2}=0, x^{2}-y^{2}-z^{2}+1=0\).

### Short Answer

Expert verified

Components of X are constrained by a circle and additional conditions on x.

## Step by step solution

01

## Analyze the given equations

Consider the two equations defining the set\[x^{2} + y^{2} + z^{2} = 0\]\[x^{2} - y^{2} - z^{2} + 1 = 0\].

02

## Subtract the equations to simplify

Subtract the second equation from the first:\[(x^{2} + y^{2} + z^{2}) - (x^{2} - y^{2} - z^{2} + 1) = 0\]This yields:\[x^{2} + y^{2} + z^{2} - x^{2} + y^{2} + z^{2} - 1 = 0\]Simplify to get:\[2y^{2} + 2z^{2} - 1 = 0\]or\[y^{2} + z^{2} = \frac{1}{2}\].

03

## Express in terms of new variable

Let \(a = \sqrt{2}y\) and \(b = \sqrt{2}z\). Then the equation becomes:\[a^{2} + b^{2} = 1\].

04

## Substitute back for x

Replace \(y\) and \(z\) back in the first equation:\[x^{2} + \frac{a^{2}}{2} + \frac{b^{2}}{2} = 0\].

05

## Identify irreducible components

Since \(a^{2} + b^{2} = 1\) from the substitution is a circle, and \(x\) must satisfy \(x^{2} = -\frac{1}{2}\), the irreducible components of set \(X\) are characterized by this constraint.

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### irreducible components

In algebraic geometry, an irreducible component of a set is a subvariety that cannot be divided into simpler subvarieties within the parent variety. When we decompose a set into its irreducible components, we are essentially breaking it down into its simplest building blocks.

For the given exercise, we have two polynomial equations defining a closed set in affine space \(\(\mathbb{A}^{3}\)\). By simplifying these equations, we derive simpler equations like \(\(y^{2} + z^{2} = \frac{1}{2}\)\). Rewriting this with new variables \(\(a = \sqrt{2}y\)\) and \(\(b = \sqrt{2}z\)\), transforms it into \(\(a^{2} + b^{2} = 1\)\), representing a circle in the \(\(a,b\)\) plane.

Next, substituting these back into the initial equations, the resulting equation for x gives \(\(x^{2} = -\frac{1}{2}\)\). Since there is no real number solution to this equation, the x-component could be interpreted over a field with different characteristics or complexes.

Understanding these components helps us to see that the set defined by the original equations can be visualized as the intersection of simpler geometric shapes like circles.

###### projective space

Projective space is a fundamental concept in algebraic geometry that extends the ordinary notion of space by adding 'points at infinity.' These are crucial for understanding how geometric objects behave under perspective transformations.

Typically denoted as \(\(\mathbb{P}^{n}\)\), projective space allows us to treat lines that may seem parallel in affine space (like \(\(\mathbb{A}^{3}\)\)) as if they intersect at a point at infinity. This is particularly useful in algebraic geometry because it simplifies the understanding of intersection properties of curves and surfaces.

In the provided problem, while the solution is confined within affine space, extending this understanding into projective space offers further insight. For example, considering the solution in \(\(\mathbb{P}^{3}\)\) may reveal additional geometric patterns and properties.

###### field characteristic

The characteristic of a field is a crucial concept in algebra and algebraic geometry, impacting how equations and algebraic structures behave.

The field characteristic is the smallest number of times the multiplicative identity (1) must be added to itself to get zero. If no such number exists, the field is said to have characteristic zero. Examples include the fields of rational, real, and complex numbers.

In the provided exercise, the field characteristic is specifically stated to be not equal to 2. This is significant because it affects the simplicity and forms of certain mathematical operations.

Most notably, had the characteristic been 2, our manipulations (including evolving the equations and their simplifications) would have been different. In characteristic not 2 fields, we can safely perform divisions by 2, allowing us to rewrite \(\(y^{2} + z^{2} = \frac{1}{2}\)\) effectively.

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