I am beginner at the model theory and I learn compactness theorem at the class and I saw some application of it and one of them is that "every field has an algebraic closure". How can I prove it with compactness theorem ?
2026-04-08 18:37:16.1775673436
Model theoretic answer for having algebraic closure
321 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in FIELD-THEORY
- Square classes of a real closed field
- Question about existence of Galois extension
- Proving addition is associative in $\mathbb{R}$
- Two minor questions about a transcendental number over $\Bbb Q$
- Is it possible for an infinite field that does not contain a subfield isomorphic to $\Bbb Q$?
- Proving that the fraction field of a $k[x,y]/(f)$ is isomorphic to $k(t)$
- Finding a generator of GF(16)*
- Operator notation for arbitrary fields
- Studying the $F[x]/\langle p(x)\rangle$ when $p(x)$ is any degree.
- Proof of normal basis theorem for finite fields
Related Questions in MODEL-THEORY
- What is the definition of 'constructible group'?
- Translate into first order logic: "$a, b, c$ are the lengths of the sides of a triangle"
- Existence of indiscernible set in model equivalent to another indiscernible set
- A ring embeds in a field iff every finitely generated sub-ring does it
- Graph with a vertex of infinite degree elementary equiv. with a graph with vertices of arbitrarily large finite degree
- What would be the function to make a formula false?
- Sufficient condition for isomorphism of $L$-structures when $L$ is relational
- Show that PA can prove the pigeon-hole principle
- Decidability and "truth value"
- Prove or disprove: $\exists x \forall y \,\,\varphi \models \forall y \exists x \,\ \varphi$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Given a field $F$, start by using the compactness theorem to show that there exists a field extension $F^*$ of $F$ such that every nonconstant polynomial in $F[x]$ has a root in $F^*$. (When applying compactness, you should be considering a theory over a language that includes constants for the elements of $F$.)
What happens when you iterate this process?