📖 Group Homomorphism A function $\\phi: G \\to H$ between groups is a \\textbf{homomorphism} if $\\phi(ab) = \\phi(a)\\phi(b)$ for all $a, b \\in G$. It is an \\textbf{isomorphism} if it is also a bijection. From: df-course Learn more:

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📖 Bijective A function is bijective if it is both injective and surjective. Bijections have inverses. From: Real Analysis Learn more:

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A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

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💡 Proposition III.18 If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent. From: Euclid's Elements Learn more:

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💡 Proposition I.26 (ASA/AAS Congruence) If two triangles have the two angles equal to two angles respectively, and one side equal to one side, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle. From: Euclid's Elements Learn more:

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📖 Discrete Valuation Ring A \\textbf{discrete valuation ring} (DVR) is a local PID that is not a field. Equivalently, it is a local ring with a unique nonzero prime ideal. From: df-course Learn more:

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📖 Discriminant of a Form The discriminant of the form $(a, b, c)$ is $D = b^2 - 4ac$. Forms with $D < 0$ are positive definite (if $a > 0$) or negative definite. From: Disquisitiones Arithmeticae Learn more:

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📖 Residue If $a \\equiv b \\pmod{m}$, then $b$ is called a residue of $a$ modulo $m$. The least non-negative residue is the unique $r$ with $0 \\le r < m$ and $a \\equiv r \\pmod{m}$. From: Disquisitiones Arithmeticae Learn more:

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📐 Properties of Minimal Polynomial The minimal polynomial $m_\\alpha$: (1) Is irreducible over $F$; (2) Divides every polynomial having $\\alpha$ as a root; (3) Has $\\deg(m_\\alpha) = [F(\\alpha):F]$. From: df-course Learn more:

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📐 Classification of Finite Fields (1) A finite field has order $p^n$ for some prime $p$ and positive integer $n$. (2) For each prime power $q = p^n$, there exists a unique (up to isomorphism) field of order $q$, denoted $\\mathbb{F}_q$ or $GF(q)$. From: df-course Learn more:

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📖 Inner Product (Dot Product) The inner product of $\\mathbf{u} = (u_1, \\ldots, u_n)$ and $\\mathbf{v} = (v_1, \\ldots, v_n)$ is $\\mathbf{u} \\cdot \\mathbf{v} = u_1v_1 + u_2v_2 + \\cdots + u_nv_n$. From: Four Pillars of Geometry Learn more:

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📖 Cycle Notation A cycle $(a_1 \\, a_2 \\, \\cdots \\, a_k)$ maps $a_i \\mapsto a_{i+1}$ (indices mod $k$). Disjoint cycles commute. From: intro-discrete Learn more:

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🎮 Interactive: Sphere Rotation Group Demo Explore the group SO(3) of all rotations in 3D space. Every rotation is a composition of two reflections! From: Four Pillars of Geometry Try it:

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Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📐 Dimension is Well-Defined Any two bases of a finite-dimensional vector space have the same number of elements. Proof: Let $\\{v_1, \\ldots, v_m\\}$ and $\\{w_1, \\ldots, w_n\\}$ be bases of $V$. Since $\\{v_1, \\ldots, v_m\\}$ spans $V$ and $\\{w_1, \\ldots, w_n\\}$ is linearly independent, by the Steinitz Exchange Lemma, $n \\leq m$. Similarly, since $\\{w_1, \\ldots, w_n\\}$ spans $V$ and $\\{v_1, \\ldots, v... From: Advanced Linear Algebra Learn more:

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📐 Three Reflections Theorem Every isometry of the plane is a composition of at most three reflections. Proof: An isometry is determined by its effect on three non-collinear points. Reflection in the perpendicular bisector of $PP'$ sends $P$ to $P'$. Using at most three such reflections, we can map any triangle to its image. Since the isometry and our composition of reflections agree on three non-collinea... From: Four Pillars of Geometry Learn more:

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💡 Proposition VI.8 (Right Triangle Altitude) If in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another. From: Euclid's Elements Learn more:

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🎮 Interactive: Derivative Visualizer See how the derivative measures instantaneous rate of change. Watch the tangent line move along any curve. From: Calculus I Try it:

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📐 Eisenstein For $f \\in \\mathbb{Z}[x]$, if prime $p$ divides all but the leading coefficient, does not divide the leading, and $p^2$ does not divide the constant, then $f$ is irreducible over $\\mathbb{Q}$. From: intro-discrete Learn more:

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📐 Chinese Remainder Theorem If $m_1, m_2, \\ldots, m_k$ are pairwise coprime, then the system $x \\equiv a_i \\pmod{m_i}$ for $i = 1, \\ldots, k$ has a unique solution modulo $M = m_1 m_2 \\cdots m_k$. Proof: For each $i$, let $M_i = M/m_i$. Since $\\gcd(M_i, m_i) = 1$, there exists $y_i$ with $M_i y_i \\equiv 1 \\pmod{m_i}$. Then $x = \\sum_{i=1}^k a_i M_i y_i$ satisfies all congruences. Uniqueness follows from the fact that any two solutions differ by a multiple of $M$. From: Disquisitiones Arithmeticae Learn more:

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💡 Proposition II.6 If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line. From: Euclid's Elements Learn more:

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💡 Proposition II.12 (Law of Cosines, Obtuse) In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. From: Euclid's Elements Learn more:

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