📖 Transcendence Degree A transcendence basis of $E/K$ is a maximal algebraically independent subset $B \\subseteq E$ over $K$. The transcendence degree $\\mathrm{tr.deg}(E/K) = |B|$ is an invariant of the extension (independent of the choice of $B$). From: gal-jacobson Learn more: https://mathacademy-cyan.vercel.app/#/section/20 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Theorem 3 The row (column) vector space $F^n$ of all $n$-tuples from a field $F$ is a vector space of dimension $n$ over $F$. Proof: The standard basis vectors $\\epsilon_1 = (1,0,\\ldots,0), \\ldots, \\epsilon_n = (0,\\ldots,0,1)$ are independent and generate $F^n$, so the dimension is $n$. From: gal-artin Learn more: https://mathacademy-cyan.vercel.app/#/section/3 Explore all courses: https://mathacademy-cyan.vercel.app

💡 Proposition (Reduction by a Cyclic Extension) Consider the Galois group of $f(x) = 0$ over a field $K$. Let $p$ be a prime, let $K$ contain primitive $p$th roots of unity, and let $K\ Proof: Let $H(X)$ be the irreducible factor of $F(X)$ over $K\ From: gal-edwards Learn more: https://mathacademy-cyan.vercel.app/#/section/14 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Classical Impossibility Results It is impossible to (1) double the cube ($\\sqrt[3]{2}$ is not constructible since $[\\mathbb{Q}(\\sqrt[3]{2}):\\mathbb{Q}] = 3$), (2) trisect a general angle, or (3) square the circle ($\\pi$ is transcendental). From: gal-morandi Learn more: https://mathacademy-cyan.vercel.app/#/section/14 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Sufficient Condition for Diagonalizability If $T \\in \\mathcal{L}(V)$ has $\\dim V$ distinct eigenvalues, then $T$ is diagonalizable. From: linalg-axler Learn more: https://mathacademy-cyan.vercel.app/#/section/15 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Algebraic Element An element $\\alpha \\in E$ is algebraic over $F$ if it is a root of some nonzero polynomial in $F[X]$. Otherwise, $\\alpha$ is transcendental over $F$. From: gal-weintraub Learn more: https://mathacademy-cyan.vercel.app/#/section/4 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Theorem 9 (Existence of Splitting Fields) If $p(x)$ is a polynomial in a field $F$, there exists a splitting field $E$ of $p(x)$. Proof: Factor $p(x)$ into irreducible factors. If any has degree {%RECENT_POSTS%}gt; 1$, adjoin a root via Kronecker\ From: gal-artin Learn more: https://mathacademy-cyan.vercel.app/#/section/9 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Galois Resolvent Let $f(x) = 0$ be a polynomial equation of degree $n$ with distinct roots $a, b, c, \\ldots$. A Galois resolvent is a polynomial $t = \\phi(a, b, c, \\ldots)$ in the roots with the property that it takes $n!$ different values when the roots are permuted in all possible ways. The only permutation that leaves $t$ unchanged is the identity. From: gal-edwards Learn more: https://mathacademy-cyan.vercel.app/#/section/10 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Galois Let $f(x) = 0$ be an equation with distinct roots whose Galois group over $K$ is $G$. Then $f(x) = 0$ can be solved by radicals if and only if $G$ is solvable -- that is, has a composition series $G \\supset G_1 \\supset G_2 \\supset \\cdots \\supset G_\\nu = \\{e\\}$ in which each $G_i$ is a normal subgroup of prime index in its predecessor. Proof: Necessity: If solvable by radicals, the tower of field extensions reduces the Galois group at each step to a normal subgroup of prime index (by the proposition of \u00a744). Taking only steps where the group decreases gives the composition series. Sufficiency: If $G$ is solvable, the proposition ... From: gal-edwards Learn more: https://mathacademy-cyan.vercel.app/#/section/16 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Complex Numbers A complex number is an ordered pair $(a, b)$ where $a, b \\in \\mathbf{R}$, written $a + bi$, with $i^2 = -1$. From: linalg-axler Learn more: https://mathacademy-cyan.vercel.app/#/section/0 Explore all courses: https://mathacademy-cyan.vercel.app

📐 Existence of Eigenvalues Every operator on a finite-dimensional nonzero complex vector space has an eigenvalue. From: linalg-axler Learn more: https://mathacademy-cyan.vercel.app/#/section/14 Explore all courses: https://mathacademy-cyan.vercel.app

📖 Field Extension A field extension $E/K$ is a pair of fields with $K \\subseteq E$, where $E$ inherits the field operations from $K$. The degree $[E:K]$ is the dimension of $E$ as a $K$-vector space. From: gal-jacobson Learn more: https://mathacademy-cyan.vercel.app/#/section/0 Explore all courses: https://mathacademy-cyan.vercel.app