Commit: 8ad996e2 - Tagged all DiskMat notes, revamped A&D tags and note types
Author: lhorva <lhorva@student.ethz.ch>
Date: 2025-12-10T16:53:07+01:00
Changes: 88 note(s) changed (4 added, 84 modified, 0 deleted)
ℹ️ Cosmetic Changes Hidden: 84 note(s) had formatting-only changes and are not shown below
AR?8CyMux0
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What is a polynomial over a commutative ring?
A polynomial \(a(x)\) over a commutative ring \(R\) in the indeterminate \(x\) is a formal expression of the form: \[ a(x) = {{c2::a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1 x + a_0}} = \sum_{i=0}^d a_i x^i \] for some non-negative integer \(d\), with \(a_i \in R\).
The set of polynomials in \(x\) over \(R\) is denoted \(R[x]\).
What is a polynomial over a commutative ring?
A polynomial \(a(x)\) over a commutative ring \(R\) in the indeterminate \(x\) is a formal expression of the form: \[ a(x) = {{c2::a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1 x + a_0}} = \sum_{i=0}^d a_i x^i \] for some non-negative integer \(d\), with \(a_i \in R\).
The set of polynomials in \(x\) over \(R\) is denoted \(R[x]\).
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>What is a polynomial over a commutative ring?</p> <h1>Back</h1> <p>A polynomial \(a(x)\) over a commutative ring \(R\) in the indeterminate \(x\) is a formal expression of the form: \[ a(x) = {{c2::a_d x^d + a_{d-1}x^{d-1} + \cdots + a_1 x + a_0}} = \sum_{i=0}^d a_i x^i \] for some non-negative integer \(d\), with \(a_i \in R\).</p> <p>The set of polynomials in \(x\) over \(R\) is denoted {{c4::\(R[x]\)}}.</p> |
lq:b}[Y<9t
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Lagrange Interpolation for polynomials in a Field
Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\).
Then \(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{{{c2::(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}}}{{{c3::(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}}}\]
Note that for \(u_i(x)\) to be well-defined, all constant terms \(\alpha_i - \alpha_j\) in the denominator must be invertible. This is guaranteed in a field since \(a_i - a_j \neq 0\) for \(i \neq j\) (as they are all distinct).
Lagrange Interpolation for polynomials in a Field
Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\).
Then \(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{{{c2::(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}}}{{{c3::(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}}}\]
Note that for \(u_i(x)\) to be well-defined, all constant terms \(\alpha_i - \alpha_j\) in the denominator must be invertible. This is guaranteed in a field since \(a_i - a_j \neq 0\) for \(i \neq j\) (as they are all distinct).
| Field | Before | After |
|---|---|---|
| Text | <h1>Front</h1> <p>Lagrange Interpolation for polynomials in a Field</p> <h1>Back</h1> <p>Let \(\beta_i = a(\alpha_i)\) for \(i = 1, \dots, d+1\).</p> <p>Then \(a(x)\) is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial \(u_i(x)\) is: \[u_i(x) = \frac{{{c2::(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}}}{{{c3::(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}}}\]</p> <p>Note that for \(u_i(x)\) to be well-defined, all constant terms \(\alpha_i - \alpha_j\) in the denominator must be invertible. This is guaranteed in a field since \(a_i - a_j \neq 0\) for \(i \neq j\) (as they are all distinct).</p> |
y`5($Q$d37
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| Field | Before | After |
|---|---|---|
| Text | {{c1::Eine Gruppe}} ist eine {{c1::Menge \(G\) mit Operation \( * \)}} mit folgenden Eigenschaften:<ul>{{c2::<li> Assoziativität: \((a * b) * c = a * (b*c)\)</li><li>Neutrales Element existiert: \( a * e = e * a = a \)</li><li>Jedes Element \(a\in G\) hat eine Inverse: \( a * a^{-1} = a^{-1} * a = e\)</li>}}<br></ul> |
BG+yKyLb,^
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| Field | Before | After |
|---|---|---|
| Text | {{c1::Ein Körper}} ist eine Menge {{c1::\( \mathbb{K}\) mit Operationen \(+ , *\)}} mit folgenden Eigenschaften:<div>{{c2::<div>- \( (\mathbb{K}, +)\) ist eine abelsche Gruppe</div><div>- \( (\mathbb{K} \backslash \{0\}, *)\) ist eine abelsche Gruppe</div><div>- Distributivität: \( a * (b+c) = a*b + a*c\)</div>}}<br></div> | |
| Extra | Beispiel: \( \mathbb{Q}, \mathbb{R}\) |