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 ...

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5
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1answer
179 views

Determining A Splitting Field

I am trying to determine the splitting fields of a bunch of polynomials. I'll ask one here and hope that a general enough technique can be described to find the rest of them. Currently, I'm trying ...
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2answers
206 views

Algebraic Field Extensions

This is exercise $6.34$ from Lang's book: Give an example of a field $K$ which is of degree $2$ over two distinct subfields $E$ and $F$, respectively, but such that $K$ is not algebraic over ...
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2answers
244 views

Application of the Artin-Schreier Theorem

This is exercise $6.29$ out of Lang's book: Let $K$ be a cyclic extension of a field $F$, with Galois group $G$ generated by $\sigma$. Assume that the characteristic is $p$, and that ...
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2answers
196 views

how many elements are there in this field

$\mathbb Z_2[x]/\langle x^3+x^2+1\rangle $, I understand it is a field as $\langle x^3+x^2+1\rangle $ ideal is maximal ideal as the polynomial is irreducible over $Z_2$. but I want to know how many ...
0
votes
1answer
79 views

How can one go from “no roots” to “is irreducible” in this case? [duplicate]

This problem is paraphrased from an old version of an exam that I will be taking, and I have no idea how one would do solve it. Let $p$ be a prime number, let $F$ be a field of characteristic ...
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0answers
60 views

Minimal Polynomials of numbers from trigonometry

What is the minimal polynomial over $\Bbb Q[x]$ of $$\alpha[k] = n\cdot ...
5
votes
1answer
112 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
1
vote
1answer
107 views

Minimal Polynomial Trouble

Let $L$ be an extension of a field $F$. Let $\alpha_1, \alpha_2\in L$ be such that both of them are algebraic over $F$ and have the same minimal polynomial $m$ over $F$. We know that there is an ...
5
votes
3answers
737 views

What is the field $\mathbb{Q}(\pi)$?

I'm having a hard time understanding section 29,30,31 of Fraleigh. In 29.16 example, what is the field $\mathbb{Q}(\pi)$? and why is it isomorphic to the field $\mathbb{Q}(x)$ of rational functions ...
5
votes
1answer
98 views

Element of with n0n-zero trace

Let $F$ be a field of characteristic $p$ and $K$ a finite, separable extension of $F$ such that $p \mid [K : F]$. I want to show that there must exist an element of $K$ with non-zero trace. One idea ...
5
votes
2answers
129 views

Can I embed $\Bbb{C}(x)$ into $\Bbb{C}$?

Is it possible to embed $\Bbb{C}(x)$ (the field of rational functions over the compelx numbers) in $\Bbb{C}$ ?
4
votes
2answers
212 views

Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$

Every field contains an isomorphic copy of the prime field $\mathbb{Q}$ if char $= 0$, and $\mathbb{Z}_p$ if char $= p$. Could you help me prove this theorem? My professor introduced this ...
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vote
1answer
74 views

Discriminant of $f(x)$ is in the ideal of $\mathbb{Z}[x]$ generated by $f(x)$ and $f'(x)$

I am reading Rational Points on Elliptic Curves by Silverman and Tate, and on page 48 authors make the following remark: (bold emphasis mine) If $f(x)$ is any polynomial with leading coefficient ...
2
votes
2answers
193 views

Is it true that $p\mid [\mathbb Q(\cos{\frac{\pi}{p^2}}):\mathbb Q]$?

Is it true that $$p\mid [\mathbb Q(\cos{\frac{\pi}{p^2}}):\mathbb Q],\forall p\in \mathbb P,$$ where $[K:\mathbb Q]$ is the degree of a field extension ?
5
votes
2answers
553 views

Determine the degree of the splitting field for $f(x)=x^{15}-1$.

Let $f(x)= x^{15} - 1$. Let $L$ be the splitting field of $f(x)$ over the field $K$. Determine the extension degree $[L:K]$ in each case. a) $K= \Bbb{R}$ b) $K= \Bbb{Q}$ c) $K = ...
6
votes
1answer
2k views

Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm) [duplicate]

Let $K$ be an arbitrary field, $p$ a prime and $a\in K$. Show $f=x^p-a$ is either irreducible in $K[x]$ or has a root in $K$. My strategy was to split this up into a case for each ...
2
votes
0answers
93 views

Necessary criterion for a field extension to be normal

I'm working on a lemma concerning some Galois theory and arithmetics. Let $p$ be an odd prime and $K/F$ be a finite Galois extension of number fields of order prime to $p$ with Galois group $H$. Let ...
3
votes
2answers
269 views

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$ for some $n$, where $\zeta_n=e^{2\pi i/n}$? I think the answer is no, but I can't give a full proof. Assume the contrary, we then have ...
2
votes
1answer
214 views

Understanding Dummit & Foote p.528 Sec 13.2 Algebraic Extensions

I can't understand why the authors conclude Hence, the elements $\alpha_i\beta_j$ span the composite extension $K_1K_2$ over $F$. I would like to understand what the authors mean and how they ...
13
votes
4answers
463 views

A field of order $32$

I was working on this problem from an old qual exam and here is the question. In particular this is not for homework. True or False: There are no fields of order 32. Justify your answer. ...
3
votes
2answers
65 views

Understanding of a theorem about criterion for multiple zeros

This is from Gallian's Contemporary Abstract Algebra: I don't understand the step in the red box: why can we assume that the common factor has a zero? Isn't it possible that for example $f(x)$ and ...
1
vote
1answer
77 views

Computation of the Wirtinger derivative of a product (continuation)

Let us have a real function $f=(A/2)\phi\bar{\phi}$, where $\phi, \bar\phi$ are complex fields. When looking for the stationary state of this function, we can either treat $\phi, \bar\phi$ as ...
2
votes
1answer
120 views

Characterization of normal and separable extensions in terms of embeddings

Let $E/F$ be a finite field extension. Let $\text{Emb}(E/F)$ denote the set of field homomorphisms $E \to \overline{F}$ that fix $F$. Here, $\overline{F}$ is the algebraic closure of $F$. My ...
2
votes
2answers
46 views

fields containing roots

Let $f(x) = x^3 - 2 \in \mathbb{Q}[x]$. Then $f(x)$ has roots $\sqrt[3]{2}$, $\sqrt[3]{2}\omega$, and $\sqrt[3]{2}\omega^2$, where $\omega$ is a complex cube root of $1$. In Birkhoff and Maclane's ...
7
votes
6answers
746 views

Are all finite fields isomorphic to $\mathbb{F}_p$?

I've recently started taking some algebra courses and I was wondering whether or not every finite field is isomorphic to $\mathbb{F}_p$, where $p$ is prime.
2
votes
1answer
206 views

Dummit Foote Exercise 13.3.15

A Field F is said to be formally real if $-1$ can not be written as sum of squares in F. let $f(x)$ be an irreducible polynomial in $F[x]$ of odd degree with $\alpha$ as a root. Now the Question is ...
1
vote
1answer
23 views

Let $\tau\in \operatorname{Aut}_K F$ and $H$ be an intermediate field. Must $\tau[H]\subseteq H$

Let $F$ be an extension field of $K$. Let $\tau\in \operatorname{Aut}_K F$ and $H$ be an intermediate field. Must $\tau[H]\subseteq H$ ? In case $\dim_K H\leq2$, the answer is easily yes. Thank you ...
2
votes
1answer
266 views

Inseparable finite extensions a field with non zero characteristic.

Suppose K is a field with characteristic p which is not a perfect field. Then how do we prove that there does exist an irreducible polynomial which is not separable. I have no idea how to proceed ...
1
vote
1answer
74 views

Splitting field over $K$ of a finite set of polynomials

$F$ is a splitting field over $K$ of a finite set $\left\{ f_{1},\dots,f_{n}\right\} $ of polynomials in $K\left[x\right]$ if and only if $F$ is a splitting field over $K$ of the single polynomial ...
4
votes
1answer
623 views

Separable closure of a field

Could someone tell me the definition of the separable closure of a field $K$ and whether it is a Galois extension of $K$? Also why is this construction useful? Many thanks
3
votes
1answer
2k views

Construct a finite field of 16 elements and find a generator for its multiplicative group.

Construct a finite field of 16 elements and find a generator for its multiplicative group. Find all generators of multiplicative group. Very obvious Construction of a field with 16 elements according ...
2
votes
1answer
57 views

An advanced algebra question

How can I show that $\Bbb{R}(x)$ (the quotient field of $\Bbb{R}[x]$) is not a real closed field ?
1
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1answer
69 views

Type $\sim$ Minimal Polynomial & Orbit

In Model Theory by Wilfrid Hodges, he gives an intuition of what a type is in the following way: "One can think of types as a common generalisation of two well-known mathematical notions: the ...
2
votes
1answer
100 views

extension of field of arbitrary degree

Is the following statement true? Let $F$ be a field and $n \in \mathbb{N}$, then there exists an extension $E$ of $F$ with $[E:F]=n$. I'm thinking of $[\mathbb{C}:\mathbb{R}]=2$ and since I can ...
4
votes
1answer
138 views

Field Extensions of degree $2$

I know that if $K$ is an extension of $\mathbb{Q}$ of degree $2$, then $K = \mathbb{Q}(\sqrt{d})$ for some squarefree integer $d$. I understand that this is not the case for degree $2$ extensions of ...
1
vote
3answers
102 views

Where are extension elements in a field extension taken from?

For example, how can you conclude that a finite field extension of an algebraic field $K$ is algebraic if you don't know the type of the extension elements (the elements that you add to extend the ...
5
votes
1answer
127 views

Quadratic closure in characteristic 2

Let $F$ be a field of characteristic 2. I need to show the existence of a quadratic polynomial in $F[t]$ which cannot be solved by adjoining all square roots of elements in the field. Attempt: For ...
5
votes
1answer
205 views

Computing the degree of a Galois extension.

Let $K = \Bbb{Q}(3^{1/5}, 3^{1/3})$. Compute $[K: \Bbb{Q}]$ (degree of the extention). Find $\alpha$ (not unique) such that $F=\Bbb{Q}(3^{1/5}, 3^{1/3} , \alpha)$ is the smallest Galois extension ...
5
votes
1answer
580 views

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$. My justification for this question is as follows; Suppose $F(\alpha^2)\subsetneq F(\alpha)$, we have $F \subsetneq F(\alpha^2) ...
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vote
0answers
57 views

Categoricity of the real numbers

Assume we have a field $R \subset \mathbb{R}$ such that $\mathrm{card}(R) = 2^{\aleph_0}$. Under what conditions it follows that $R=\mathbb{R}$?
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votes
1answer
91 views

Number of elements of a subset of $\mathbb F_8$

I wish somebody could help me in this. I encountered this question in a previous year paper of an exam. Let $F$ be a field with $8$ elements and $A=\{x\in F\mid x^7=1$ and $x^k \ne 1$ for all ...
5
votes
3answers
538 views

Show that $f(x) = x^p -x -1 \in \Bbb{F}_p[x]$ is irreducible over $\Bbb{F}_p$ for every $p$.

a) Show that $f$ has no roots in $\Bbb{F}_p$ Let $F^*$ be the multiplicative group of $\Bbb{F}_p$. Then, by lagrange's thoerem for all nonzero $a \in \Bbb{F}_p$, $x^{p-1} = 1 \implies x^p=x ...
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votes
1answer
309 views

Isn't $(0)$ a prime ideal in a field?

I have read in multiple places that a field $K$ has a Krull dimension of $0$. How is this true? Isn't $(0)\subset K$ a prime ideal in $K$? Obviously $K$ is an integral domain. Thanks in advance!
1
vote
1answer
129 views

Problem about intermediate fields in the extension

Let $E,K$ be intermediate fields in the extension $L/F$ (a) If $[EK:F]$ is finite, then $$[EK:F] \leq [E:F][K:F] $$ (b) If $E$ and $K$ are algebraic over $F$, then so is $EK$ For (a), I try two ...
6
votes
2answers
265 views

Extension of residue fields and algebraic independence

Let $A$ be a Noetherian integral domain, $B$ a ring extension of $A$ that is an integral domain, $P \in \operatorname{Spec} B, \, p = P \cap A$. Denote by $\kappa(p),\ \kappa(P)$ the residue fields of ...
49
votes
12answers
5k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
3
votes
3answers
132 views

isomorphic polynomial rings

I'm certain that this is a dumb question, but I'll ask anyway. I know that if $\theta : F \to K$ is a field isomorphism then we get an induced isomorphism $\varphi:F[x] \to K[x]$ such that $\varphi|F ...
2
votes
1answer
31 views

Are composite fields unique?

Suppose for $i=1,2$ that $\Omega_i$ is a field containing fields $K_i$ and $L_i$, with $K_1 \cong K_2$ and $L_1 \cong L_2$. Is it then true that there is an isomorphism $K_1L_1 \cong K_2L_2$ of ...
6
votes
2answers
280 views

Polynomial rings — Inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
2
votes
1answer
61 views

Extensions of number fields

Let $L|K$ be an extension of number fields and consider the corresponding (integral) extension of ring of integers: $R_L|R_K$. Note that $R_L$ and $R_K$ are finitely generated over $\mathbb{Z}$, hence ...