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content/Chinese Remainder Theorem.md

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+---
+id: 1725891281-chinese-remainder-theorem
+aliases:
+  - Chinese Remainder Theorem
+tags: []
+---
+
+# Chinese Remainder Theorem
+## Modular Arithmetic Basics:
+- When we say $a \equiv b \text{ (mod m)}$, we mean that $a$ and $b$ have the same remainder when divided by $m$.
+- "$m$ divides $(a-b)$"
+
+## The Problem CRT Solves:
+- A system of simultaneous congruences.
+- Given a set of congruences: $x \equiv a_1 \text{ (mod } m_1 \text{)} \quad x \equiv a_2 \text{ (mod } m_2 \text{)} \dots x \equiv a_n \text{ (mod } m_n \text{)}$
+- Where the moduli ($m_1,m_2,\dots,m_n$) are pairwise [[coprime]], CRT states that there exists a unique solution modulo $M=m_1*m_2*\dots *m_n$
+
+## Constructing the Solution:
+- Calculate $M=m_1*m_2*\dots *m_n$
+- For each $i$, calculate $M_i=M/m_i c$
+- Find the modular multiplicative inverse of $M_i$ modulo $m_i$, call it $y_i$ ($M_i y_i \equiv 1 \text{ (mod } m_i \text{)}$)
+- The solution is: $x \equiv (a_1M_1y_1+a_2M_2y_2+\dots+a_nM_ny_n)$ (mod $M$).
+
+## Why It Works
+- Key insight: $M_i$ is divisible by all moduli except $m_i$
+- When we multiply it by $y_i$, we get a number that is congruent to $1$ modulo $m_i$ and congruent to $0$ modulo all other moduli
+- This allows us to "select" the right $a_i$ for each modulus.

content/Extension fields.md

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+---
+id: 1725898288-extension-fields
+aliases:
+  - Extension fields
+tags: []
+---
+
+# Extension fields
+Fundamental concept in abstract algebra, esp field theory. Crucial role in understanding structure of fields and have important applications in areas like algebraic number theory and [[Galois theory]].
+
+## Definition
+An extension field is a larger field that contains a given base field as a subfield. If $F$ is the base field and $E$ is the extension field, we denote this as $E/F$ ("E over F").
+
+## Degree of Extension
+Degree of an extension $E/F$, denoted $[E:F]$, is the dimension of $E$ as a vector space over $F$. It can be finite or infinite.
+
+## Types of Extensions
+- Simple Extension
+    - An extension $E/F$ is simple if $E=F(\alpha)$ for some $\alpha \in E$
+    - $\alpha$ is called a primitive element of the extension
+- Algebraic Extension
+    - An extension $E/F$ is algebraic if every element of $E$ is algebraic over $F$ (i.e., is the root of some polynomial with coefficients in $F$)
+- Transcendental Extension
+    - An extension that is not algebraic
+    - Contains at least one element that is not the root of any polynomial over $F$
+- Finite Expression
+    - An extension with finite degree
+    - All finite extensions are algebraic
+- Normal Extension
+    - An algebraic extension where every irreducible polynomial over $F$ that has one root in $E$ has all its roots in $E$
+- Separable Extension
+    - An algebraic extension where the minimal polynomial of every element is separable (has no repeated roots in its splitting field)
+- Galois Extension
+    - An extension that is both normal and separable
+
+## Constructing Extension Fields
+- Adjoining Roots
+    - Given a polynomial $f(x)$ over $F$, we can construct an extension field by adjoining a root of $f(x)$ to $F$
+- Splitting Fields
+    - The smallest extension field that contains all roots of a given polynomial
+
+## Algebraic Closure
+- An algebraically closed field is one in which every non-constant polynomial has a root
+- Every field has an algebraic closure, which is an algebraic extension that is algebraically closed
+
+## Tower Law
+- For extensions $E/K$ and $K/F$, we have $[E:F]=[E:K][K:F]$.
+- This is crucial in analysing the structure of field extensions.
+
+## Applications
+- Solving Polynomial Equations
+    - Extension fields allow us to find roots of polynomials that don't have roots in the base field
+- Constructibility Problems
+    - In geometry, extension fields help determine which lengths are constructible with compass and straightedge
+- [[Galois Theory]]
+    - Uses field extensions to study the solvability of polynomial equations and the relationship between roots of polynomials
+- Algebraic Number Theory
+    - Studies extensions of the rational numbers
+    - Crucial in understanding properties of algebraic numbers
+
+## Primitive Element Theorem
+- Any finite separable extension has a primitive element, meaning it can be expressed as a simple extension
+
+## Fundamental Theorem of [[1725899101-galois-theory|Galois Theory]]
+- Establishes a correspondence between intermediate fields of a Galois extension and subgroups of its Galois group
+- Provides a powerful tool for analysing field extensions

content/Fast Fourier Transform.md

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+---
+id: 1725903044-fast-fourier-transform
+aliases:
+  - Fast Fourier Transform
+tags: []
+---
+
+# Fast Fourier Transform
+Highly efficient algorithm for computing the Discrete Fourier Transform (DFT) of a sequence. It's a fundamental tool in signal processing.
+
+## Basic Concept
+- FFT rapidly computes the DFT by factoring the DFT matrix into a product of sparse (mostly zero) factors
+- Reduces the complexity from $O(N^2)$ for the naive DFT implementation to $O(N\log N)$, where $N$ is the length of the input sequence
+
+## Key Ideas
+- Divide and Conquer approach
+- Exploits symmetries and periodicities in the complex [[roots of unity]]
+- Recursively breaks down a DFT of size $N$ into smaller DFTs
+
+## Cooley-Tukey Algorithm
+- Most common FFT algorithm
+    - Radix-2 Decimation in Time (DIT)
+        - Splits the input sequence into even and odd-indexed sub-sequences
+        - Recursively computes the DFT of these sub-sequences
+        - Combines results using the [[butterfly operation]]
+    - Radix-2 Decimation in Frequency (DIF)
+        - Similar to DIT, but splits the output spectrum instead of the input sequence
+
+## Other FFT Algorithms
+- Split-Radix FFT: more efficient for power-of-two sizes
+- Bluestein's Algorithm: handles arbitrary input sizes
+- Rader's algorithm: efficient for prime-sized inputs
+
+## Complexity
+- Time complexity: $O(N\log N)$
+- Space complexity: $O(N)$ for in-place algorithms
+
+## Key Properties
+- *Linearity* — $\text{FFT}(ax+by)=a\text{FFT}(x)+b\text{FFT}(y)$
+- *Parseval's theorem* — energy conservation between time and frequency domains
+- Convolution theorem — convolution in time domain equals multiplication in frequency domain
+
+## Applications
+- Digital signal processing
+- Audio and image compression (e.g., MP3, JPEG)
+- Polynomial multiplication
+- Fast multiplication of large integers
+- Solving partial differential equations
+- Spectral analysis in various scientific fields
+
+## Inverse FFT
+- Inverse transform can be computed using the same algorithm with slight modifications
+- Maintains $O(N\log N)$ complexity
+
+## Simple Python Implementation of Cooley-Tukey
+```python
+import cmath
+
+def fft(x):
+    N = len(x)
+    if N <= 1:
+        return x
+    even = fft(x[0::2])
+    odd = fft(x[1::2])
+    T = [cmath.exp(-2j * cmath.pi * k / N) * odd[k] for k in range(N // 2)]
+    return [even[k] + T[k] for k in range(N // 2)] + \
+           [even[k] - T[k] for k in range(N // 2)]
+
+# Example usage
+x = [1, 2, 3, 4, 5, 6, 7, 8]  # Input should have a length that is a power of 2
+X = fft(x)
+print("Input:", x)
+print("FFT:", [f"{z.real:.2f} + {z.imag:.2f}j" for z in X])
+
+# Verify by comparing with numpy's FFT
+import numpy as np
+np_fft = np.fft.fft(x)
+print("NumPy FFT:", [f"{z.real:.2f} + {z.imag:.2f}j" for z in np_fft])
+```
+
+## Key points about the FFT
+- Twiddle Factors: The complex exponentials $(e^{(-2πik/N)})$ are often called twiddle factors.
+- Bit-Reversal: Many FFT implementations use a technique called bit-reversal to reorder input or output elements.
+- Real-valued FFT: When the input is real-valued (as in many practical applications), specialized algorithms can nearly halve the computation time.
+- Pruning: When many input values are zero or many output values are not needed, pruning techniques can speed up the computation.
+- Parallel Implementations: The FFT algorithm is well-suited for parallel computing, including GPU implementations.

content/Galois Theory.md

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+---
+id: 1725899101-galois-theory
+aliases:
+  - Galois Theory
+tags: []
+---
+
+# Galois Theory
+Provides a deep connection between field theory and group theory.
+
+## Core Idea
+- Establishes correspondence between subfields of a [[1725898288-extension-fields|field extension]] and the subgroups of a certain group of automorphisms of the extension field
+- This correspondence allows us to study the structure of field extensions using group theory
+
+## Galois Group
+- For a field extension $E/F$, the Galois group $\text{Gal}(E/F)$ is the group of all [[1725899418-automorphism|automorphisms]] of $E$ that fix every element of $F$
+- These automorphisms permute the roots of polynomials over $F$ in $E$
+
+## Galois Extensions
+- An algebraic extension $E/F$ is called a Galois extension if it is both normal and separable
+- For such extensions, the Galois theory correspondence is most complete
+
+## Fundamental Theorem of Galois Theory
+This is the cornerstone of Galois theory. For a finite Galois extension $E/F$:
+- There is a one-to-one correspondence between the intermediate fields $K$ $(F ⊆ K ⊆ E)$ and the subgroups $H$ of $\text{Gal}(E/F)$.
+- If $K$ corresponds to $H$, then $[E:K] = |H|$ and $[K:F] = [\text{Gal}(E/F):H]$.
+- $K$ is the fixed field of $H$ (elements of $E$ fixed by all automorphisms in $H$).
+- $\text{Gal}(E/K) = H$.
+- $K$ is a normal extension of $F$ if and only if $H$ is a normal subgroup of $\text{Gal}(E/F)$.
+
+## Applications
+- Solvability of Polynomial Equations:
+    - Galois theory provides a criterion for determining whether a polynomial equation is solvable by radicals. An equation is solvable by radicals if and only if its Galois group is a solvable group.
+- Impossibility Proofs:
+    - It proves the impossibility of certain geometric constructions (like trisecting an angle or squaring a circle) and the non-existence of formulas for roots of general polynomials of degree 5 or higher.
+- Field Automorphisms:
+    - Helps in understanding the structure of field automorphisms, crucial in algebraic number theory and algebraic geometry.
+
+## Steps in Galois Theory Analysis
+1. Identify the splitting field of a polynomial.
+2. Determine the Galois group of this splitting field over the base field.
+3. Analyse the subgroups of the Galois group.
+4. Use the Galois correspondence to understand the structure of subfields.
+
+## Advanced Concepts
+- Inverse Galois Problem:
+    - Asks which finite groups can be realized as Galois groups over a given field. Still open for the rational numbers.
+- Infinite Galois Theory:
+    - Extends the ideas to infinite extensions, using topological groups.
+- Differential Galois Theory:
+    - Applies Galois-theoretic ideas to differential equations.
+
+## Historical Significance
+- Galois theory revolutionized algebra by shifting focus from finding explicit solutions to understanding the structure of the solutions.
+- It laid the groundwork for much of modern abstract algebra.
+
+## Connection to Other Areas
+- Galois theory has profound connections to number theory, algebraic geometry, and even areas of physics like quantum mechanics.

content/Polynomial Arithmetic.md

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+---
+id: 1725898229-polynomial-arithmetic
+aliases:
+  - Polynomial Arithmetic
+tags: []
+---
+
+# Polynomial Arithmetic
+## Addition and Subtraction
+Add or subtract the coefficients of like terms.
+
+Example: $(3x² + 2x + 1) + (x² - x + 4) = 4x² + x + 5$
+
+## Multiplication
+Use the distributive property and combine like terms.
+
+Example: $(x + 2)(x - 3) = x² - 3x + 2x - 6 = x² - x - 6$
+
+## Division
+Long division or synthetic division for dividing by linear factors.

content/Polynomials.md

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+---
+id: 1725897202-polynomials
+aliases:
+  - Polynomials
+tags: []
+---
+
+# Polynomials
+An expression consisting of variables (usually denoted by letters) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
+
+General form: $P(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n$ where $a_0,a_1,\dots,a_n$ are constants (coefficients) and $n$ is a non-negative integer.
+
+## Key Terms
+- *Degree* — the highest power of the variable in the polynomial
+- *Leading coefficient* — the coefficient of the highest degree term
+- *Constant term* — the term without a variable ($a_0$ in the general form)
+- *Monomials* — polynomials with only one term
+
+## [[Polynomial Arithmetic]]
+### Addition and Subtraction
+Add or subtract the coefficients of like terms.
+
+Example: $(3x² + 2x + 1) + (x² - x + 4) = 4x² + x + 5$
+
+### Multiplication
+Use the distributive property and combine like terms.
+
+Example: $(x + 2)(x - 3) = x² - 3x + 2x - 6 = x² - x - 6$
+
+### Division
+Long division or synthetic division for dividing by linear factors.
+
+## Important Theorems:
+### Fundamental Theorem of Algebra:
+Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
+
+### Factor Theorem:
+A polynomial $P(x)$ has a factor $(x - a)$ if and only if $P(a) = 0$.
+
+### Remainder Theorem:
+The remainder of $P(x)$ divided by $(x - a)$ is equal to $P(a)$.
+
+## Roots and Factoring
+Roots (or zeros) are values of $x$ that make $P(x) = 0$.
+Factoring is expressing a polynomial as a product of simpler polynomials.
+
+Methods include:
+- Factoring out common factors
+- Grouping
+- Difference of squares: $a² - b² = (a+b)(a-b)$
+- Sum/difference of cubes: $a³ ± b³ = (a ± b)(a² ∓ ab + b²)$
+
+## Polynomial Functions
+A polynomial function is a function defined by a polynomial.
+
+### Properties
+- Continuous everywhere
+- Differentiable everywhere
+- The degree determines the maximum number of turning points and x-intercepts
+
+## Applications
+- Curve fitting and data approximation
+- Signal processing
+- Control systems
+- Cryptography (e.g., in error-correcting codes)
+
+## Advanced Concepts
+- [[Polynomial Rings]]
+    - Algebraic structures consisting of all polynomials over a given field.
+- [[Galois Theory]]
+    - Uses properties of polynomials to study field extensions and solvability of equations.
+- Orthogonal Polynomials
+    - Special polynomial sequences used in mathematical analysis and physics.
+
+## Computational Aspects
+- Evaluating polynomials efficiently (e.g., Horner's method)
+- Finding roots numerically (e.g., Newton's method)
+- Interpolation: constructing a polynomial that passes through given points