0
votes
1answer
30 views

What exactly are the elements of $\mathbb{Z}_p[x]/\langle p(x) \rangle$?

It is wellknown that for a polynomial ring $\mathbb{Z}_p[x]$, $\mathbb{Z}_p[x]/\langle p(x) \rangle$ for prime $p$ is a field if and only if $p(x)$ is irreducible over the given polynomial ring, in ...
2
votes
1answer
78 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$‚ÄČ to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
0
votes
0answers
30 views

Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
3
votes
1answer
38 views

Number of isomorphisms between two fields

Let $F,F'$ be two fields. Is there anything that can be said about the number of isomorphisms that can exist? In particular can there be more than one? What if $F$ is the complex numbers $\mathbb C$? ...
1
vote
2answers
70 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
2
votes
2answers
40 views

Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
0
votes
3answers
29 views

Ring theory question.

Why is a field with 27 elements has characteristic 3? I was solving a question and I came to know this fact which I didn't know before. Is there anyone who can explain this to me? Thanks in advance. ...
2
votes
2answers
75 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
3
votes
2answers
92 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. ...
0
votes
0answers
21 views

For which $d\in\mathbb{Z}$, $\mathbb{Q}(\sqrt{d})$ primitive root of unity of order $p>2$ prime

If $p>2$ is a prime number, then I have to find $d\in\mathbb{Z}$ such that we have a primitive root of unity of order $p$. I know that $d<0$ because otherwise, you can never have a root of ...
5
votes
1answer
140 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
1
vote
0answers
11 views

Why can an ideal generated by a subset be written in this form?

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative ...
2
votes
2answers
47 views

Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
0
votes
1answer
42 views

Definition of a field homomorphism

Given a field $F$ of characteristic zero, say $F=\mathbb{R}$, what is the minimal requirement for a function $\mu:F\to F$ to be a field homomorphism? (Do we need to require two axioms, one for ...
0
votes
0answers
40 views

Why is this not trivial? - $f(x)~|~g(x) \iff g(x) \in \langle f(x) \rangle$

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
2
votes
0answers
23 views

Fields and quotient ring

Let $P(X)\in{\mathbb{R}[X]}$ irreducible polynomial. Then $\mathbb{R}[X]/(P(X)=X^2+1)\approx{\mathbb{C}}$. If $P(X)=X^2+X+1$ also $\mathbb{R}[X]/(P(X))\approx{\mathbb{C}}$? Or for a arbitrary ...
0
votes
2answers
44 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
5
votes
2answers
70 views

Let $F$ be a field of order $2^n$. Prove that characteristic of $F$ is 2.

I figure that Lagrange's theorem and the fact that the characteristic of an integral domain is either $0$ or prime should be used, but just can't figure it out exactly.
-2
votes
2answers
64 views

{0} is unique maximal ideal when F is field [duplicate]

Let $R$ be a ring. Show that R is a field if and only if $\{ 0 \}$ is the unique maximal ideal of $R$. Thank you
2
votes
2answers
53 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
1
vote
1answer
40 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
0
votes
2answers
37 views

Dimension of $K\subset L(\alpha)$ where $L$ is a field extension of $K$

Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a ...
2
votes
0answers
38 views

Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
0
votes
1answer
54 views

Why principal ideal should be commutative?

According to the definition of Principal Ideal it should be commutative. What if the ring is not commutative? Which means $ar\neq ra$ where $a\in I, r \in R$. Does it lead to a contradiction? Because ...
2
votes
1answer
95 views

If $f$ is an irreducible rational polynomial then all the roots over $\mathbb{C}$ are distinct

I'm trying to show that if $f \in \mathbb{Q}[t]$ is irreducible then all the roots of $f$ in $\mathbb{C}$ are distinct. My first issue, am I right in thinking that the roots are distinct iff ...
0
votes
1answer
26 views

Ideal of a field

Let $F$ be a field. Show that $S$ be a non empty subset of $F^{n} $ then $ I(S) =$ { $ f(x) \in F[x] \hspace{0.1in} \vert \hspace{0.1in}f(s) = 0 \hspace{0.1in} \forall s \in S $ } is an ...
2
votes
1answer
40 views

Roots of polynomials in field of prime characteristic p

Let $F$ be a field of characteristic $p$, and let $\alpha \in F$. Let $f \in \mathbb{Z}_p[x]$ be such that $f(\alpha) = 0$. Apparently, we have $f(\alpha^p) = 0$. This was mentioned but not proved in ...
2
votes
1answer
111 views

Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $ g := x^n - 1 \in F[x]$ Is it true that $ g$ has distinct roots in $F$ if and only if ...
0
votes
1answer
116 views

Prove that the Gaussian rationals is the field of fractions of the Gaussian integers

I'm looking to prove that $\Bbb Q[i] = \{ p + qi : p, q \in \Bbb Q \}$ is the field of fractions of $\Bbb Z[i] = \{p + qi : p, q \in Z\}$. I am familiar with definition of a field of fractions. For ...
1
vote
1answer
56 views

Transcendental number over $\{k\in K\mid f(k)=k\}$

Let $K$ be a field and $f:K\rightarrow K$ be a ring endomorphism. Prove that if $\alpha\in K\setminus f(K)$, then $\alpha$ is transcendental over the subfield of $K$, $F:=\{k\in K\mid f(k)=k\}$. My ...
4
votes
2answers
116 views

The ring of integers of $\mathbf{Q}[i]$

Is there a relatively "simple" (in the sense that it does not require knowledge of algebraic number theory) proof that the ring of integers of the algebraic number field $\mathbf{Q}[i]$ is ...
1
vote
1answer
66 views

About the notation $\mathbb{Z}[x]/(f(x),p)$

Let $f(x)\in \mathbb{Z}[x]$ be a polynomial and $p$ be a prime. What does the notation $\mathbb{Z}[x]/(f(x),p)$ mean? Is it $\mathbb{Z}/p\mathbb{Z}[x]/(f(x))$ ?
2
votes
1answer
41 views

How to find all the roots in this ring?

Let $F:= \mathbb{F}_7[x]/(x^2+3x+1)$ Is it a field? Find all the roots in F of the polynom $f (Y) := Y^2+[3]_{F}Y +[1]_{F} \in F[Y]$. Attempt: It is a field, because $x^2+3x+1$ is irreducible ...
1
vote
0answers
34 views

kernel of maps associated to the root of an irreducible polynomial

Let $m(\mu)$ be an irreducible polynomial of degree $d$ over $\mathbb{F}_2$, $F_{2^d} = \mathbb{F}_2[x]/(m(\mu))$ by a field extension given by that polynomial and let $d: \mathbb{F}_2[x] \to ...
0
votes
1answer
30 views

Question about the quotient of two lattices.

Let $F$ be a non-archimedian local field with valuation $\nu$. Then $\mathcal{O}=\{x\in F: \nu(x)\geq 0\}$ is the ring of integers of $F$. $\mathfrak{m}=\{x\in F: \nu(x)> 0\}$ is the maximal ideal ...
2
votes
3answers
105 views

How to prove the one-variable calculus definition of derivative extends to $\Bbb C$ *only* because $\Bbb C$ is a field?

I have been told the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. See : Higher dimensional analogues of the argument principle? $$ ...
4
votes
7answers
296 views

What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?

The title pretty much says it all. Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period." In my experience, mathematical definitions ...
1
vote
1answer
64 views

Proving that set is (or is not) a field

Let $P = \{a + b\sqrt[3]3 + c\sqrt[3]9, a, b, c \in \Bbb Z \}$ It is easy to prove that $(P, +, \cdot)$ is a ring considering ordinary addition and multiplication. How to prove that this set is or is ...
-1
votes
4answers
77 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.
1
vote
3answers
55 views

Minimal polynomial of a finite field

Let $p,n\in\mathbb{N}$ with $p$ prime and $q=p^n$. Let $\mathbb{F}_q$ be the finite field with $q$ elements (unique up to $\cong$), i.e. the $q$-th Galois field. According to this, the extension ...
0
votes
3answers
94 views

About isomorphism of rings and fields

If $A,B$ are rings and $A$ is a field. If $A$ is a field and $A\cong B$ so $B$ is a field too? Thank you!
1
vote
2answers
59 views

Figuring out whether a ring is a field

Given a ring, how do you test whether it is a field? What properties would you look at?
4
votes
2answers
105 views

Showing that a ring homomorphism from a field to a ring is injective

A similar question like this has been asked here, apologies, but need to clarify something at the end Our homework question was to show that any ring homomorphism $f:K\rightarrow R $ (where K is a ...
1
vote
2answers
69 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
0
votes
1answer
36 views

For a Euclidean Domain, prove that $\delta(z)>0$ if $z\in R$ is not a unit.

For a Euclidean Domain, prove that $\delta(z)>0$ if $z\in R$ is not a unit, where $\delta$ is the Euclidean function. Is it just since $z$ is not a unit then $\delta(z)>\delta(1)>0?$ Please ...
1
vote
0answers
35 views

integral domain with a field as a subring [duplicate]

I would like to know if my solution to the following exercise is correct. Let $A$ be an integral domain (with a unit) which has a field $\mathbb K$ as a subring and such that $A$ is a ...
1
vote
2answers
199 views

Specific way of showing $\Bbb Z[\sqrt{-d}]$ is not a Euclidean Domain when $d>2$

Is it true that if a ring is not a UFD then it's not a Euclidean Domain? I have a ring $R=\mathbb{Z}[\sqrt{-d}]=\{ a+b\sqrt{-d} \mid a,b \in \mathbb{Z} \}$ where $d$ is a square free integer. I want ...
1
vote
2answers
423 views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
1
vote
1answer
65 views

What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$

I'm doing some exercises to prepare for my exam: What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$. I've no idea how to tackle this ...
7
votes
1answer
74 views

Short method to prove irreduicibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$?

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$? In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...