2
votes
2answers
104 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
0
votes
2answers
41 views

prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.

let $R$ be a unique factorization domain and let $F$ be its field of fractions. Prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.
4
votes
2answers
40 views

Why is an extension $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism if $\phi : F[x] \to F(a)$ is injective?

Why is $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism given $\phi : F[x] \rightarrow F(a)$ satisfy $\ker \phi = \{0\}$ ? I've been trying to figure out why $\bar \phi$ is an isomorphism, and ...
23
votes
6answers
1k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
2
votes
2answers
35 views

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field.

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field. Find necessary and sufficient condition Attempt: Since, we know that a finite ...
0
votes
1answer
37 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...
1
vote
1answer
46 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
3
votes
0answers
22 views

Proving that a field of characteristic $0$ is the field of fractions of a proper subring.

If $K$ is a field of characteristic $0$, $A$ is a subring of $K$ maximal subring of $K$ which doesn't contain $\frac{1}{2}$, and $F$ is the field of fractions of $K$, then I have proved that $K$ is ...
3
votes
0answers
28 views

Facts about quotient rings - example

I have three quotient rings: $R_1 = \frac{\mathbb{Q}[x]}{(x^2 -1)}$ $R_2 = \frac{\mathbb{Q}[x]}{(x^2 +1)}$ $R_3 = \frac{\mathbb{Q}[x]}{((x -1)^2)}$ I am trying to decide whether these are integral ...
-2
votes
3answers
37 views

Does represented ring appear to be a field? [closed]

$\mathbb{R}[x]/(x^{2}+1,x^3-2x^2+x-2)$ Hello! My name is Ramzan! I`m solving this issue!
2
votes
0answers
36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
1
vote
2answers
47 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
0
votes
2answers
30 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
2
votes
1answer
42 views

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$.

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$. Here $K$ is UFD, $Q$ is its field of fractions, $Q(\beta)$ is algebraic extension of $n>1$ degree of ...
2
votes
1answer
65 views

Show that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic to the field $\mathbb{Q}$

I have problem in showing that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic as a ring to the field $\mathbb{Q}$. Any idea? Thanks
1
vote
2answers
47 views

Integral domain (rings and fields)

Let $\mathbb Z[i]=\{a+ib \mid a, b \in \mathbb Z \}$. How to Show that $\mathbb Z[i]$ is a integral domain?
1
vote
1answer
47 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks
1
vote
3answers
128 views

Questions about $\Bbb Q[\sqrt{p}]$ and $\Bbb Q(\sqrt{p})$

I studied this part where they talk about $\Bbb{Q}(\sqrt{2})$ and $\Bbb{Q}[\sqrt{2}]$ and I really start to get confused. Definitions: $$ \Bbb{Q}[\sqrt{2}] = \left\{ a + b \sqrt{2} \mid a,b \in ...
1
vote
0answers
38 views

Showing that $\frac{\mathbb{C}[X]}{<x-1>}$ is isomorphic to $\mathbb{C}$

I'm trying to show that $\frac{\mathbb{C}[X]}{<x-1>} \cong \mathbb{C}$ and I am not sure if this argument is correct. Define $\phi: \mathbb{C}[X] \to \mathbb{C}$ by $\sum a_it^i \to \sum a_i$. ...
5
votes
0answers
29 views

Is there an online database somewhere that lists identities for algebraic structures with two binary operators?

I'm working on an abstract algebra library in Python, and I'm trying to include as many functions that analyze algebraic structures, returning true or false based on whether or not the algebra ...
3
votes
0answers
53 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
3
votes
2answers
81 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ...
2
votes
1answer
35 views

$\mathbb Q$ Field extension

Consider the Field $F = \mathbb Q(2^{\frac 1 3})$, Is $\sqrt 2 \in F$? I'm trying to figure out how to determine that and similar questions, can you give me a hint or some guidance on how to do that? ...
2
votes
1answer
27 views

understanding roots of polynomials in field extensions

I'm running into a conceptual stumbling block understanding the application of the FHT to field extensions and finding roots, if anyone has any pointers on where I might be misunderstanding. I'm ...
0
votes
2answers
31 views

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R-M$ is a unit. Then $R/M$ is a field.

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field. I am solving this question of NBHM 2011. To solve this is ...
2
votes
3answers
46 views

How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$

How can one deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$ ? $\mathbb Q (u) = \{c_0 +c_1u+c_2u^2 | c_i \in \mathbb Q\}$ Is it a trivial computation, or does this ...
4
votes
1answer
42 views

Why is the order of a prime element well-defined?

This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ ...
4
votes
0answers
67 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
1
vote
1answer
34 views

Prove that $B$ is a subfield of $F$

If a subring $B$ of a field $F$ is closed with respect to multiplicative inverses, then $B$ is a field. Fields are commutative rings with unity, and every nonzero element has an inverse. A subring ...
2
votes
2answers
39 views

Polynomial ring and extension field

Let $K$ be a field and $p(x) \in K[x]$ a monic irreducible polynomial of degree $n$, suppose $F/K$ is a field extension, and there exist $u \in K[x]$ which is a root of $p(x)$. 1) Let $K(u)$ be the ...
1
vote
1answer
30 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
4
votes
1answer
33 views

Finding the conjugates in $\mathbb{C}$ of a given number over a given field…

I'm having somewhat of a difficult time understand what's being asked—and thus having a hard time answering the question: Find all conjugates in $\mathbb{C}$ of the given number over the given ...
-1
votes
1answer
62 views

Irreducibility over $ \mathbb{Q} ( \sqrt{2} , \sqrt{3})$ [closed]

Show that $x^5-9 x^3 +15x +6$ is irreducible over $ \mathbb{Q} ( \sqrt{2}, \sqrt{3})$
3
votes
1answer
181 views

Online Finite Field Calculator

I need to find an online Finite Field calculator with the following operations: Inverse SqrRoot Mult Div I have found one a couple of days ago but lost the url, and cannot find it now. Any ...
0
votes
1answer
32 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
89 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  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
32 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
42 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
74 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
48 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
30 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
139 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
169 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
26 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
143 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
53 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
43 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
42 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
27 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 ...