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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
5 views

Under what condition given $(x_1, y_1\cdot r_1),…,(x_n, y_n\cdot r_n)$ we can interpolate polynomial $T$ that has specific random root?

We know given $(x_1, y_1),...,(x_n, y_n)$ we can interpolate a polynomial $P$ of of degree at most $n-1$. Let us define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is at most $n-1$, ...
1
vote
1answer
21 views

Finding ring isomorphisms

Let $A$ be a ring with $0\neq 1$ such that $x^4=1, \forall x\in A$, with $x\neq 0$. My question is: to which ring is $A$ isomorphic? $A$ can be, for example, isomorphic to $\mathbb{Z}_2$. The ...
3
votes
3answers
25 views

Let $h(x)= x^p − a^{p−1}x ∈ k[x]$. Show that $k(h)$ is the fixed field of $φ$.

Let $k$ be a field of characteristic $p>0$ ,and let $a∈k$. Let $h(x)= x^p − a^{p−1}x ∈ k[x]$. Show that $h$ is fixed by the automorphism $φ$ of $k(x)$ defined by $φ(f (x)/g(x)) = f (x + a)/g(x + ...
-3
votes
1answer
37 views

$F$ is subfield of complex field $\mathbb{C}$. Show that $F$ is field with characteristic $0$ [on hold]

We know that subfield of field is set $F$ of complex number which itself is a field under usual multiplication and addition. but how to show that it has characteristic $0$?
0
votes
1answer
20 views

field extension $F\subset E$ with both separable and inseparable elements

Can someone please give me an example of a field extension $F\subset E$ such that E\F has both separable and inseparable elements? if $F(\alpha)$ is a simple extension of F, and if $\alpha$ is ...
1
vote
0answers
17 views

Finite ring having $2^n-1$ invertible elements [duplicate]

Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ ...
-1
votes
2answers
37 views

Field and Field Axioms.

I wanted to ask what are field and field axioms? I have tried looking on Wikipedia and Wolfram But They are too are advanced and I cant a understand one bit.So please any help would be much ...
0
votes
1answer
51 views

Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible [duplicate]

Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible using this hint: If $a$ is a root of $X^p-X-1$, show that $a^{p^{p}}=a$, who is the extension ...
2
votes
0answers
23 views

Show that $K=k(u)$ iff $u=\frac{ax+b}{cx+d}$ where $a,b,c,d \in k$ with $ad-bc \neq 0$

Let $k$ be a field and let $K=k(x)$ be the rational function field in one variable over $k$. If $u\in K$, show that $K=k(u)$ iff $u=\frac{ax+b}{cx+d}$ where $a,b,c,d \in k$ with $ad-bc \neq 0$. ...
1
vote
1answer
62 views

Fixed Field of $\sigma, \tau$

Let $k$ be a field and let $K=k(x)$ be the rational function field in one variable over $k$. Let $\DeclareMathOperator{\aut}{Aut}\sigma, \tau \in \aut(K)$ s.t. $$\sigma\left(\frac {f(x)}{ ...
0
votes
0answers
27 views

what is the minimal condition for two elements to create same field extension?

Given a field $K\subset E$, with $\alpha,\beta\in E$, such that $K(\alpha)=K(\beta)$. What can we then say about $\alpha$ and $\beta$? If the extension is finite, then $\alpha$ is a linear ...
2
votes
0answers
35 views

Find $\operatorname{min}(\omega\sqrt{2},\mathbb{Q})$ [duplicate]

Let $\omega$ be a primitive third root of unity with $K=\mathbb{Q}(\omega,\sqrt{2})$. Find $\operatorname{min}(\omega\sqrt{2},\mathbb{Q})$ Could anyone tell me how to find this? and generally which ...
2
votes
2answers
43 views

Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$

I want to show rigorously that factorization over algebraic extensions of $\mathbb{Q}$ automatically yields a corresponding factorization over $\mathbb{F}_p.$ Consider for example the polynomial ...
0
votes
1answer
36 views

Find elements $x,y$ where $x\ne \pm1$ and $y\ne \pm 1$ in the field $\mathbb{Q}(\sqrt{5})$ satisfying $xy=19$.

Find elements $x,y$ where $x\ne \pm1$ and $y\ne \pm 1$ in the field $\mathbb{Q}(\sqrt{5})$ satisfying $xy=19$. I'm lost as to what to do. Any solutions or hints are greatly appreciated.
0
votes
1answer
26 views

Check if algebraic structure is a field

Check if algebraic structure $(\mathbb{R^2},+,\cdot)$ is a field where binary operations $(+)$ and $(\cdot)$ are given by $$(x,y)+(u,v)=(x+u,y+v)$$ $$(x,y)\cdot(u,v)=(xu-2yv,xv+yu)$$ Structure ...
0
votes
2answers
23 views

Let $q$ be a prime integer. Show that for each $x∈GF(q)$ there exist elements $r$ and $s$ in $GF(q)$ satisfying $x=r^2+s^2$.

Let $q$ be a prime integer. Show that for each $x∈GF(q)$ there exist elements $r$ and $s$ in $GF(q)$ satisfying $x=r^2+s^2$. I'm stuck on this problem. Any solutions or hints are greatly appreciated. ...
4
votes
1answer
135 views

Splitting field as a terminal object?

Let $f(x)\in K[x]$ be a polynomial over field $K$ and let $E$ be a splitting field. I would like to prove that $E$ is unique up to isomorphism by expressing the inclusion $K\to E$ as a terminal object ...
0
votes
0answers
40 views

Is the set containing just zero a mathematical field? [duplicate]

Consider the set $\left\lbrace0\right\rbrace$ together with the usual operations of addition and multiplication. Is this set together with these operations a field? I know that one of the ...
2
votes
1answer
20 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
1
vote
0answers
32 views

Equivalence of two statements in a field.

I need to prove the following in order to prove something interesting about generalized quaternions: Let $K$ be a field and suppose $a \neq 0, b \neq 0$ are elements of $K$. Then the following are ...
4
votes
1answer
32 views

Find a $u$ so that $k(u)$ is the fixed field of $φ$, determine the minimal polynomial over $k(u)$

Let $k = F_p$, and let $k(x)$ be the rational function field in one variable over $k$. Define $φ : k(x) \to k(x)$ by $φ(x) = x+1$. Show that $φ$ has finite order in $Gal(k(x)/k)$. Determine this ...
0
votes
0answers
13 views

for which $\alpha \in K$ $det(xI-L_{\alpha})=min(F,\alpha)$

Let $K $ and $F$ are two fields, $K=F(a)$ suppose $[K:F]=n$ for $\alpha \in K$ let $L_{\alpha}$ be the $F$ linear transformation $K$ to $K$ defined by $L_{\alpha}(x)=\alpha x$. Now my question is for ...
0
votes
1answer
26 views

A problem regarding field theory

I came across this problem in N.Jacobson's 'Basic Algebra' (Vol I): Let $E = (\mathbb{Z}/(p))(t)$, where $t$ is trasncendental over $\mathbb{Z}/(p)$. Let $G$ be the group of automorphisms generated ...
0
votes
0answers
30 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
1
vote
0answers
27 views

What's so special about quadratic extensions?

Reading through chapter 13 "Field Theory" from Dummit and Foote Algebra. I am wondering why such an emphasis is placed upon "quadratic extensions" of a field F. They state that for any field F ...
0
votes
0answers
31 views

Complete field and field extension.

$(K,u)$ be a pair of the field $K$ and its absolute value $u$, $(K_u, \bar u)$ denotes its completion and the corresponding absolute value. Let $L$ be a field containing $K$, $\pi:K_u\rightarrow ...
1
vote
1answer
43 views

Quadratic field extensions and complex conjugation

If you consider any quadratic extension $K$ of $\mathbb{Q}$, it has to be fixed by complex conjugation, because from $[K : \mathbb{Q}] = 2$ we know $K | \mathbb{Q}$ has to be a normal extension and as ...
0
votes
0answers
32 views

What are complex and real dimensions of this space?

Show that the vectors $v_1$ = (i,1+i,2+i), $v_2$ = (1,1+i,2+i), and $v_3$ = (2,-i,-i) form a basis for the complex vector space $C^3$.... Show that $v_1,iv_1,v_2,iv_2,v_3,iv_3$ is a basis of ...
2
votes
0answers
17 views

Proving that the set of separable elements over a field is a field itself.

My field theory book says that the set of separable elements over a field is a field itself. This roightly translates to the fact that of $a $ and $b $ are separable, so are $a+b, ab, 1/a$, etc. I ...
5
votes
3answers
81 views

Understanding $Gal(\bar k /k)$

According to this book, An Introduction to the Langlands Program, One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a ...
0
votes
1answer
32 views

Minimal polynomial is irreducible

Suppose $\mathbb{E}$ is a field extension of $\mathbb{F}$. If $a$ is algebraic over $\mathbb{F}$ we define the minimal polynomial for $a$ as the monic irreducible generator $g$ of the ideal ...
0
votes
0answers
27 views

Square root in a general field

In $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ there are obvious ways to calculate the square root of a quadratic residue. For finite fields of order $p$ we can use the Tonelli–Shanks algorithm. How ...
4
votes
1answer
38 views

For which $n \in \mathbb{N}$ $f(x) = x^{2n}+x^n+1$ is irreducible in $\mathbb{F}_2[x]$?

I have $$f(x) = x^{2n}+x^n+1 \in \mathbb{F}_2[x].$$ When is this polynomial irreducible? It is obvious that for even $n$ this polynomial is reducible. But I don't have any idea about odd $n$.
2
votes
3answers
72 views

Existence of solution in finite field .

Show that a solution always exists for $X^2+Y^2 = -1$ in any finite field $\mathbb{Z}_p$. For $p$'s of the form $4n +1$, it's easy to prove that by taking $Y =0$. I couldn't figure out how to tackle ...
0
votes
0answers
28 views

Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive

Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive. Prove also that even though the Galois group of $f$ is transitive not every permutation ...
3
votes
1answer
75 views

Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$?

Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? Can we construct that proper subring? Is it necessarily an integral domain?
0
votes
1answer
31 views

If $E/F$ is finite divisible, then it is separable.

I want build a separable extension $E/F$. Suppose that $E/F$ is a finite divisible field extension. I want to prove that $E/F$ is separable in this method: we know that if $Char(F)=0$, then $E/F$ is ...
3
votes
1answer
39 views

Fraction field of $F[X,Y](f)$ isomorphic to $F(X)[Y]/(f)$

Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$. Then, by Gauss's lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field ...
2
votes
3answers
73 views

Does $\cos (\pi/5)$ belong to $\mathbb{Q} (\sin(\pi/5))$?

I need to know if $$\cos(\pi/5) \in \mathbb{Q} (\sin(\pi/5))?$$ I can compute explicitly such $\cos$ and $\sin$, but I have some difficulties how to deduce from this an answer.
2
votes
2answers
27 views

Can a field have isomorphic but unequal finite extension?

Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. Then, does there exists finite extension $E_1,E_2$ of $\mathbb{Q}$ inside $\overline{\mathbb{Q}}$, such that $E_1\neq E_2$ but ...
0
votes
0answers
13 views

Multiplicative Identity as a proof

Is there an underlying, more primitive or basic branch of Mathematics that can be used to prove the Field Axiom: 1*A = A instead of having to accept it as an unproven Axiom?
2
votes
2answers
17 views

Irreducibility criteria for polynomials with several variables.

Let $K$ be a field. Show that $x^2-yz$ is irreducible in $K[x,y,z]$. Deduce that $x^2-yz$ is prime. If it is $K[x]$, then there are several methods which can be used to check whether a given ...
0
votes
0answers
15 views

What is the characteristic of fields quasi real-closed fields?

Suppose that $F$ is a field such that $F^*$ is isomorphic to the multiplicative group of a real-closed field. I researched about these fields and I found that they can have $Char(F)=p>2$, but it ...
3
votes
1answer
134 views

Galois group of a characteristic polynomial

Quick question. I have a 3x3 matrix with integer entries, say $J$. $J$ has rank 2 and determinant equal to zero. According to GAP the matrix has characteristic polynomial $f = X^3-9X^2+8X$. Moreover ...
-3
votes
1answer
47 views

Prove that $R$ is also a field. [closed]

Let $K$ be an algebraic extension of $F$ and let $R$ be a subring of $K$ and $F \subseteq R \subseteq K$. Then prove that $R$ is also a field.
0
votes
1answer
23 views

Prove that $E=F[\alpha^2]$ [duplicate]

Let $E=F[\alpha]$, $\alpha$ is algebraic over $F$ and $[E:F]$ is odd. Prove that $E=F[\alpha^2]$. Now clearly $[F[\alpha^2]:F]|[E:F]$ so $[F[\alpha^2]:F]$ is also odd. But how can we show that ...
0
votes
1answer
55 views

Adjoining all roots of unity to an arbitrary field $F$, is an abelian extension?

I want to build an abelian extension of an arbitrary field $F$, such that any polynomial $x^n-1$ for all $n\in \mathbb{N}$ has an answer in it. So I want to adjoin all roots of unity to $F$. But I ...
1
vote
1answer
28 views

A question regarding proving the fact that every finite field is perfect

I am trying to prove the fact that every finite field is perfect. Hence, every irreducible polynomial is separable (does not have a repeated root). This is easy to prove when in a field of ...
9
votes
3answers
120 views

When is a Morphism between Curves a Galois Extension of Function Fields

My apologies if this question has already been answered somewhere on this site: when I searched, I could only find specific examples rather than the general question. It is known that the category of ...
1
vote
0answers
18 views

determinant of independent set of triangles

let $n > 3$. To a square free trinomial $x_i x_j x_k$ associate the $n$ vector that has all entries zeros except in the $i$-th, $j$-th, and $k$-th entries, where it has the value $1$ (i.e. all ...