📖 Linear Independence A list $v_1, \\ldots, v_m$ in $V$ is linearly independent if the only choice of $a_1, \\ldots, a_m \\in \\mathbf{F}$ that makes $a_1 v_1 + \\cdots + a_m v_m = 0$ is $a_1 = \\cdots = a_m = 0$. From: linalg-axler Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Galois A polynomial $f(x) \\in K[x]$ (with $\\mathrm{char}(K) = 0$) is solvable by radicals if and only if its Galois group $\\mathrm{Gal}(f)$ is a solvable group. From: gal-jacobson Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Irreducible Element (in a Ring) An element $p$ of a ring is irreducible if it is not a unit (not $\\pm 1$ in $\\mathbb{Z}$) and if the only factorizations $p = ab$ have one of $a$, $b$ as a unit. In $\\mathbb{Z}$, irreducible elements are the prime numbers and their negatives. From: gal-edwards Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Theorem 6 (Tower Law) If $F \\subset B \\subset E$ are three fields, then $(E/F) = (B/F)(E/B)$. Proof: If $A_1, \\ldots, A_r$ are independent over $B$ and $C_1, \\ldots, C_s$ are independent over $F$, then the $rs$ products $C_i A_j$ are independent over $F$. A dependence relation $\\sum a_{ij} C_i A_j = 0$ forces $\\sum a_{ij} C_i = 0$ for each $j$ (by independence of $A_j$ over $B$), which force... From: gal-artin Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Generalized Eigenspace Decomposition If $V$ is a complex vector space and $T \\in \\mathcal{L}(V)$ has eigenvalues $\\lambda_1, \\ldots, \\lambda_m$, then $V = G(\\lambda_1, T) \\oplus \\cdots \\oplus G(\\lambda_m, T)$. From: linalg-axler Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Theorem 7 (Kronecker) If $f(x)$ is a polynomial in a field $F$, there exists an extension field $E$ of $F$ in which $f(x)$ has a root. Proof: Factor $f(x)$ into irreducible factors. For an irreducible factor of degree $n$, construct $E_1 = F[\\xi]/(f(\\xi))$, the set of formal polynomials in a symbol $\\xi$ of degree $< n$, with multiplication defined modulo $f(\\xi)$. The irreducibility of $f$ guarantees that $E_1$ is a field in which... From: gal-artin Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Divisibility of Polynomials If $f(x) = g(x) \\cdot h(x)$, then $g(x)$ divides $f(x)$ in $F$, or equivalently, $g(x)$ is a factor of $f(x)$. From: gal-artin Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Structure of Simple Algebraic Extensions If $\\alpha$ is algebraic over $K$ with minimal polynomial $f(x)$ of degree $n$, then $K(\\alpha) \\cong K[x]/(f(x))$ and $[K(\\alpha):K] = n$. A basis for $K(\\alpha)/K$ is $\\{1, \\alpha, \\alpha^2, \\ldots, \\alpha^{n-1}\\}$. From: gal-jacobson Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Inner Product An inner product on $V$ is a function $\\langle \\cdot, \\cdot \\rangle: V \\times V \\to \\mathbf{F}$ satisfying positivity, definiteness, additivity in the first slot, homogeneity in the first slot, and conjugate symmetry. From: linalg-axler Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Solvable Group A group $G$ is said to be solvable if there exists a sequence of subgroups $G = G_0 \\supset G_1 \\supset G_2 \\supset \\cdots \\supset G_\\nu = \\{e\\}$ in which each $G_i$ is a normal subgroup of $G_{i-1}$ of prime index, and the final subgroup $G_\\nu$ contains only the identity. Such a sequence is called a composition series (with prime factors). From: gal-edwards Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Solution to Noether A system $\\{x_\\sigma\\}$ indexed by elements of a group $G$ of automorphisms of $E$ is a solution to Noether\ From: gal-artin Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 L\u00FCroth If $K \\subset L \\subset K(x)$ with $L \\neq K$ (where $x$ is transcendental over $K$), then $L = K(u)$ for some $u = f(x)/g(x) \\in K(x)$. Every intermediate field of a simple transcendental extension is itself simple transcendental. From: gal-jacobson Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

🏆 Magic Internet Math - Daily Top 10 🥇 Ape Mithrandir - 4,649 XP | Level 10 🥈 Phundamentals - 3,602 XP | Level 9 🥉 Marc - 10 XP | Level 1 4. Derek Ross - 10 XP | Level 1 5. bassload@tutanota.com - 0 XP | Level 1 6. eau - 0 XP | Level 1 7. negr0 - 0 XP | Level 1 8. Anonymous - 0 XP | Level 1 9. Anonymous - 0 XP | Level 1 10. Anonymous - 0 XP | Level 1 📚 Start learning:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Characterization of Cyclic Extensions Let $F$ contain a primitive $n$th root of unity. Then $K/F$ is cyclic of degree $n$ if and only if $K = F(\\alpha)$ where $\\alpha^n \\in F$. From: gal-morandi Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Normal Extension A finite extension $E/F$ is normal if every irreducible polynomial in $F[X]$ having a root in $E$ splits completely in $E[X]$. From: gal-weintraub Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Field Extension If $F$ and $K$ are fields with $F \\subseteq K$, then $K$ is a **field extension** of $F$, written $K/F$. The **degree** $[K:F]$ is the dimension of $K$ as an $F$-vector space. From: gal-morandi Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Theorem 2 In any generating system, the maximum number of independent vectors is equal to the dimension of the vector space. Proof: Let $A_1, \\ldots, A_r$ be a maximal independent subset of the generating system. Every remaining generator is a linear combination of $A_1, \\ldots, A_r$. If $B_1, \\ldots, B_t$ are any vectors with $t > r$, expressing each $B_j$ in terms of the $A_i$ gives a homogeneous system with more unknown... From: gal-artin Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Cubic Resolvent The resolvent cubic of a quartic $x^4 + bx^2 + cx + d$ is $y^3 - by^2 - 4dy + (4bd - c^2)$. The Galois group of the quartic is determined by the factorization of the resolvent cubic and the discriminant. From: gal-morandi Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Galois Group of the General Equation Let $K\ From: gal-edwards Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Fundamental Theorem of Linear Maps If $V$ is finite-dimensional and $T \\in \\mathcal{L}(V, W)$, then $\\operatorname{range} T$ is finite-dimensional and $\\dim V = \\dim \\operatorname{null} T + \\dim \\operatorname{range} T$. Proof: Let $u_1, \\ldots, u_m$ be a basis of $\\operatorname{null} T$. Extend to a basis $u_1, \\ldots, u_m, v_1, \\ldots, v_n$ of $V$. Then $Tv_1, \\ldots, Tv_n$ is a basis of $\\operatorname{range} T$, giving $\\dim V = m + n = \\dim \\operatorname{null} T + \\dim \\operatorname{range} T$. From: linalg-axler Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

Explore all courses:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.