I understand that the splitting field for a given polynomial is the smallest subfield of $\mathbb{C}$ for which the polynomial factors linearly. But if we know that every polynomial splits over $\mathbb{C}$, what is the purpose of finding the smallest field for which it splits?
2025-01-13 00:09:54.1736726994
Purpose of Splitting Fields
42 Views Asked by AnotherPerson https://math.techqa.club/user/anotherperson/detail AtRelated Questions in GALOIS-THEORY
- Understanding calculations of log/antilog tables of polynomials over finite field
- Give $3$ examples of a field extensions which are neither normal nor separable.
- Is there a normal extension $L$ such that $\mathbb Q \subset \mathbb Q(\sqrt3) \subset L$ with cyclic $\text{Gal}(L/\mathbb Q) \cong \mathbb Z_4^+$
- Show that $K \neq F(a)$ for any $a \in K$.
- Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
- What Is the Basis of the Splitting Field of $x^3 - 2$ over $\mathbb Q$?
- Is $\mathbb{Q}(\sqrt{2+\sqrt{-5}})$ normal over $\mathbb{Q}$
- Any difference working with matrices over fields?
- Let $F$ be a field of characteristic $p$. Show that $F$ is perfect field if and only if...
- Roots of $x^3-2=0$ over $\mathbb{Q}$
Related Questions in SPLITTING-FIELD
- I have to find a splitting field of $x^{6}-3$ over $\mathbb{F}_{7}$
- Purpose of Splitting Fields
- How do I factor $x^8-x$ over $\mathbb{Z}_2$?
- Proof Verification Regarding Image of $K$-homomorphisms of a Normal Extension
- Trying to find splitting fields over Q of $x^{19} -1$
- Determining whether $Q(i)(^4\sqrt 2) :Q(i)$ is a normal extension
- If $P\in K[X]$ irreducible, and if $\alpha_1,...,\alpha_n$ are his roots, does his splitting field is $K(\alpha_1,...,\alpha_n)$?
- Normal extension, why is $E/\mathbb F_p(t)$ normal?
- What is the splitting field of $(X^3-2)(X^3-3)(X^2-2)$
- Splitting field and field extension $\mathbb Q(j,\sqrt[3]2)$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- 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?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- 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
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity