# Questions tagged [field-theory]

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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### Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?

We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex conjucates: $z, \overline{z}$. ...
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### Integral closure of a field with uneven characteristic

I am studying a lof of algebra lately and i am stuck on this problem. Let $K$ be a field with uneven characteristic and $E = K(X)(\alpha)$ where $\alpha^2= X^2 -1 \in K(X)$ I am struggling to find ...
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### Find splitting field of $(x^3-x^2-x)(x^4-x^2+1)$ over $\mathbb{F}_3$

As written in the title, I have to compute the splitting field of $$(x^3-x^2-x)(x^4-x^2+1)$$ over $\mathbb{F}_3$ I'd like to understand if my attempt is correst, or if I'm missing something. Here's ...
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### Integer basis for $\mathbb{Z}_{\mathbb Q(\alpha)}$

So we are studying integral field and i am stuck on this problem. Let $f(X) = X^3 -X -12 \in \mathbb{Q}[X]$ and $\alpha \in \bar{\mathbb{Q}}$ be a zero of $f(X)$. I already showed that $f(X)$ is ...
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I have an exercise from my course notes that states: Find how many irreducible factors has $f(x) = x^{26}-1$ over $\mathbb{F}_3$ and their degrees. (don't factorize it) I see immediately that the $... 1answer 11 views ### Confusion Over Polynomial Separability Example In lecture notes I am reading for Abstract Algebra, in the Separability section for Field Theory, it gives the introduction and definition that a separable polynomial is one that has no repeated roots ... 0answers 22 views ### Convergent power series and Euclidean domain, or field. Consider the following power series (as a function of$z$): $$G(z) = \sum_{i=0}^\infty a_i z^{-i},$$ where$a_i$is a real coefficient and$zis any complex number. It also satisfies that $$\sum_{... 1answer 23 views ### Number of maximal ideal in the quotient ring [closed] How to find the number of maximal ideal in the Quotient ring \frac {\mathbb{Z}_5{[x]}}{<(x+1)^2(x+2)^3>}. ? R/A is an integral domain iff A is prime. 1answer 24 views ### Is there an isomorphism of fields between \mathbb{F}_{3^{2}} and \mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\}? if \mathbb{F}=\{a+bi; a,b \in \mathbb{F}_{3}\} where i=\sqrt{2}=\sqrt{-1} and we define (a+bi)+(c+di):=(a+c)+(b+d)i and (a+bi)\ast (c+di):=(ac-bd)+(ad+bc)i Is there an isomorphism of fields ... 1answer 27 views ### Unique subfield L of K=\mathbb{Q}(\sqrt{2},\zeta_5) such that [K:L]=5 K is the splitting field of the polynomial f=x^5-2, and we need to prove the existence of a unique subfield L such that [K:L]=5. It would be nice to use the Galois group here, but it is a semi-... 1answer 13 views ### Definition of non split Cartan subgroup What is the definition of the non split Cartan subgroups of GL_2(\mathbb{F}_p)? And what are the explicit expression of a matrix of this subgroups? I read on "Modular Functions of One Variable III" ... 3answers 57 views ### Proving F is a field [closed] Let F=\{a+bi; a,b \in F_3\} where i=\sqrt{2}=\sqrt{-1} and we define (a+bi) + (c+di) := (a+c)+(b+d)i and (a+bi) * (c+di):= (ac-bd)+(ad+bc)i with 0=0+i and 1=1+0i Prove F is a field 0answers 11 views ### Show this linear map over a finite field is not the zero map [duplicate] Let q=p^k be a prime power, let m\in\mathbb{N} and consider the map$$ \tau:\mathbb{F}_{q^m}\longrightarrow\mathbb{F}_{q^m},\quad a\longmapsto\sum_{i=0}^{m-1} a^{q^i}. $$Show that \tau is ... 4answers 40 views ### For which n is \sum_{i=0}^n x^i\in\mathbb{Q}[x] irreducible? I know that when n+1=p is prime, f=\sum_{i=0}^{p-1} x^i is the minimal polynomial of \zeta_p, a primitive p-th root of unity, hence is irreducible. This can be shown by applying Eisenstein to ... 4answers 357 views ### Understanding Fraleigh's proof of: Every finite integral domain is a field Here's how Fraleigh proves: Every finite integral domain is a field in his book: Let \begin{equation*} 0, 1, a_1, \dots, a_n \end{equation*} be all the elements of the ... 1answer 22 views ### Splitting field L of polynomial f \in K[x] with degree n satisfies [L:K] | n! Suppose f \in K[x] is a polynomial with degree n, f = (x-\alpha_1)...(x-\alpha_n) over the algebraic colsure. Let L=K(\alpha_1,...,\alpha_n) be the splitting field of f. Prove that [L:K] ... 1answer 32 views ### Degree of extension \mathbb{C}/K, where K is maximal with the property \sqrt{2} \notin K This question has been asked before but not really answered, but my query is a bit separate. To summarise the details: K is a field maximal with respect to the property \sqrt{2}\notin K, any ... 2answers 104 views ### If the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist? I have a question: if the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist? 0answers 28 views ### Degree of an exclusionary Field Extension Let's say I've got a field \mathbb{Q}[i]\ \mathbb{Q}. What's the degree of the field extension \mathbb{Q}[i]\ \mathbb{Q} : \mathbb{Q}? Clearly without the exclusion this has a degree of 2; ... 1answer 32 views ### f,g \in k[t] with k(f,g)=k(t), \deg(f)=2 and \deg(g)=3 Let f=f(t),g=g(t) \in k[t], k is a field of characteristic zero. Assume that the following two conditions are satisfied: (i) \deg(f)=2 and \deg(g)=3. (ii) k(f,g)=k(t). Question: Is it ... 0answers 31 views ### Why a polynomial is irreducible over Z Prove that the following polynomial is irreducible over \mathbb{Z}:$$f(x) = x^8-x^7+x^5-x^4+x^3-x+1$$My attempt: one can see that f(x)=(x^4+x^3+1)(x^4+x+1) over \mathbb{F_2} where these 2 ... 1answer 32 views ### Galois group of field extension I was asked to find the Galois group of the extension \mathbb{Q}(\sqrt{2},\sqrt{2},e^{\frac{2\pi i}{3}}). Since the degree of the minimal polynomials of \sqrt{2},\sqrt{2} and e^{\frac{2\pi ... 1answer 49 views ### Finding a minimal polynomial of a root of unity over a field extension I'm trying to find the minimal polynomial of the seventh root of unity over the field Q(i\sqrt{7}). I know how to do this over the rationals and have proceeded to finding that (x-1)(x^6 +x^5 +x^4 +... 1answer 73 views ### f,g \in k[t] such that \deg(f)=\deg(g) and k(f,gt)=k(t) Let f=f(t), g=g(t) \in k[t] be two nonzero polynomials over a field k of characteristic zero. Assume that the following two conditions are satisfied: (i) \deg(f)=\deg(g) \geq 1. (ii) k(f,gt)=... 1answer 28 views ### How to find generators for the subfields of \mathbb{Q}(\zeta_{12}) This is somewhat of a follow-up to this question: A complete picture of the lattice of subfields for a cyclotomic extension over \mathbb{Q}. After reading this, I am still confused on how to find ... 1answer 33 views ### Write de addition and multiplication tables for this field [closed] F3 is the set of residue class of 3 how can i do the multiply and addition table for this field of 9 elements 0answers 26 views ### Does adjoining an element to a field change the characteristic? (No it doesn't.) This may have an answer somewhere already, but I can't find it. Let K be a field of characteristic p. Let's adjoin, for example, \zeta_n where \zeta_n is some primitive n-th root of unity. ... 2answers 34 views ### (Proof verification) there is no homomorphism between a finite field’s additive group to its multiplicative group. Given a finite field F with additive group \text{F}^+ and multiplicative group \text{F}^{\times} Show that there doesn’t exist f:\text{F}^+ \to \text{F}^{\times} s.t. f(x+y)=f(x)f(y). Proof ... 1answer 41 views ### If K(\zeta,\beta)/K is the splitting field for f, what can we say about K(\zeta,\beta)/K(\zeta)? Problem setup: Let n\in\mathbb{N} and let K be a field whose characteristic does not divide n. The splitting field of f=x^n-c (c\not=0) over K is K(\zeta,\beta) where \zeta is a ... 1answer 25 views ### Equivalence between integral closures I am struggling with the following problem. Let S be a Integral domain and K its Quotient field. Furthermore let A/B/K be Field extensions. We set X = \text{Int}_S(B) where \text{Int}_S(B) ... 0answers 13 views ### Question regarding p-basis in complete local ring Let k be the residue field of characteristic p>0 of a complete local ring R with maximal ideal m. A p-basis for k is a subset B \subset k such that (i) k^p(B)=k (ii) all ... 1answer 33 views ### Finding class number of quadratic number field using Minkowski bound My understanding of this is as follows: In the general case, one has a quadratic number field F, which is always of the form \mathbb{Q}(\sqrt{d}) for some square-free integer d. Minkowski ... 1answer 29 views ### Real or imaginary part of a field extension of \mathbb{Q}. When L/\mathbb{Q} is a finite extension of \mathbb{Q} that doesn't lie in \mathbb{R}. Is there anything we can say about the degree of the extension L/ L \cap \mathbb{R}? I know that for ... 1answer 30 views ### a is a norm in K(\sqrt{a})/K if and only if it is the sum of two squares a is a norm in a quadratic extension K(\sqrt{a})/K if and only if a=u^2+v^2 for some u,v\in K. K an arbitrary field. Only if follows from: If a is a sum of squares then there exists an ... 0answers 37 views ### Find basis of \mathbb Q(\sqrt{2},\sqrt{-3}) over the field \mathbb Q(\sqrt{2}\omega). Splitting field for the polynomial \ x^3-2 over \mathbb Q is \mathbb Q(\sqrt{2},\sqrt{-3}). Now roots of the above polynomial are \sqrt{2}, \sqrt{2}\omega, \sqrt{2}{\omega}^2 Since ... 1answer 31 views ### Root of f is p^{\text{th}} power in extension field \Rightarrow coefficients of f are p^{\text{th}} powers in base field. Let K be a field of characteristic p > 0, and f ∈ K[X] monic and irreducible with root \alpha. Let F be the Frobenius endomorphism. To demonstrate: \alpha ∈ F[K(\alpha)] \Rightarrow f ∈... 3answers 67 views ### Galois group of x^p-a over \mathbb{Q} I have found that the Galois group G of f=x^p-a over \mathbb{Q} is of order p(p-1). I need to show that if P is a subgroup of G of order p, then P is normal and G/P is cyclic. ... 2answers 955 views ### Which field property enables us to multiply on both sides by the same value, while preserving equality? [duplicate] I am currently reading through Rudin's Principles of Mathematical Analysis and I am learning about fields and their properties. Note that this is the initial chapter - I am just starting off. I was ... 1answer 41 views ### Calculate order of multiplicative group of finite field How can one calculate the order of a multiplicative group of a finite field such as: (\mathbb{F}(2^3) \backslash \{0\}, \times) Is it as simple as doing 2^3-1 ? 1answer 24 views ### Is every field a field extension of some form. I am new to finite field theory .While I was going through the theory I figured that \mathbb{C} is in fact \mathbb{R}(i) isomorphic to \mathbb{R}[x]/(x^2+1) .So I had a question in mind is ... 0answers 42 views ### Can Field of fractions be Algebraically Closed? Let R be an integral domain which isn't a field. Can it be the case that the field of fractions is algebraically closed? The reason I'm asking this is : The field of fractions of \mathbb Z is \... 3answers 76 views ### Roots of x^{p^{n-1}}+\ldots+x^p+x in \mathbb{F}_{p^n} Let \mathbb{F}_q denote a field with q=p^n elements, where p is prime. Consider the polynomial f=x^{p^{n-1}}+\ldots+x^p+x and the sets$$ \begin{align*} S&=\{a^p-a:a\in\mathbb{F}_q\},\\ ... 0answers 99 views ### Galois Group ofx^{6}-2x^{3}-1$I was trying to compute the normal closure of$\mathbb{Q}[\alpha]$, where$\alpha = \sqrt{1+\sqrt{2}}$. I had a reallyyyy hard time proving that$x^{6}-2x^{3}-1$is irreducible. I proved that it ... 1answer 20 views ### Express the fixed field of a subgroup$H$of$\Gamma(L:K)$in terms of a basis of$L$over$K$. I've been trying to solve this problem for$L:K$finite with char$K = 0$:$\beta_1,...,\beta_n$basis for$L$over$K$,$H$subgroup of$\Gamma(L:K)\implies\phi(H) = K(\gamma_1,...,\gamma_n)$, ... 1answer 28 views ### Prove that$f(x)=2x^3+ax^2+bx+c$is irreducible in$ℚ[x]$if and only if$f({d\over2})≠0$for all$a, b, c, d∈ℤ$.$f(x)=2x^3+ax^2+bx+c$where$a, b, c∈ℤ$. Prove that f is irreducible in$ℚ[x]$if and only if$f({d\over2})≠0$for all$d∈ℤ$. Using a hint from my professor I have attempted a proof but I don't think ... 1answer 38 views ###$X^{p^k} - a ∈ K[X]$irreducible? Let$K$be a field of characteristic char$(K) = p > 0$, and let$a ∈ K$be an element with the following property: $$(\forall \beta ∈ K)(\beta^p ≠ a).$$ Let$k ∈ ℕ$be arbitrarily given. Is it ... 1answer 39 views ### How to show$E\otimes_k\bar k$has at least two prime ideals? [closed] Suppose$k\subsetneqq E$are two fields, and$E$is separable over$k$,$\bar k$is the algebraic closure of$k$, how to show$E\otimes_k\bar k$has at least two prime ideals? 1answer 48 views ### Product of the elements in Galois group and irreducible polynomial Statement) Let$K$be a Galois extension over a field$F \ \text{s.t.} \ \vert G(K/F) \vert =[K;F] =n $Say$ G(K/F) = \{ \sigma_1(=id) , \sigma_2,..., \sigma_n \}$Take a element of$\alpha \in K ...
When constructing a finite field $\mathrm{GF}(p^n)$ using polynomials: Why do we need to modulo an irreducible polynomial? What happens if this polynomial is reducible? Does such an irreducible ...
### Find an irreducible polynomial in $Q[x]$ of degree $726$.
Find an irreducible polynomial in $Q[x]$ of degree $726$. I first thought of $x^{726}+1$ to start but its roots would be complex so not in $Q$. Now I'm thinking to use Eisenstein, so taking the ...