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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Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$

This was an assertion made in our textbook but I have no idea how to show that either statement is true. Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, ...
1
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1answer
51 views

$\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$

Prove that $\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$, where $p$ is a prime number I know this is true for many primes that ...
1
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1answer
38 views

Field homomorphism induces an isomorphism between their prime subfields

So the question is: Let $\sigma$: $F_1 \xrightarrow[]{} F_2$ be a homomorphism where $F_1$ and $F_2$ are fields. Show $\sigma$ induces an isomorphism between their prime subfields and, in ...
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1answer
63 views

$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a \boxplus b=a+b-ab$ for all $a,b \in F$. $a \boxdot b=1-t^{\log_t ...
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2answers
28 views

Understanding the connection of the roots of an irreducible polynomial and a basis for field extensions

Let $\alpha,\beta,\gamma \in E$ be the roots of an IRREDUCIBLE polynomial $p(x)\in Q[x]$ (where E/Q is an extension field. Can I use these roots to construct a basis for E over Q? Why?
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1answer
42 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
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1answer
28 views

How to find irreducible polynomials in a given ring or field?

The other day on this site I came across a question about $\mathbb Z_3 [x]/ \langle x^2 + 1 \rangle$. At the time I misread the question and thought it was Find all the irreducible polynomials in ...
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0answers
30 views

Do addition and multiplication define a structure of a field? [duplicate]

I am taking an advanced linear algebra course for my Masters but never took linear in undergrad so please realize I know little to nothing about these topics. Question: Let r exist in R and 0 not ...
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2answers
23 views

Difficulty understanding how an element of a quotient ring/field can be represented a certain way…

This is the proposition I'm given, which I don't really understand: Let $p(x)=p_0 + p_1x + ... + p_nx^n$ be an irreducible polynomial over a field $F$, so that $ E = {f[x]}/{\lt p(x)\gt} $ is a ...
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1answer
55 views

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$.

Define multiplication as $(a,b) \boxdot (c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$ where $a,b,c,d,r\in \mathbb{R}$ and $0\neq s\in \mathbb{R}$. The multiplicative inverse is $(1,0)$. I need to show that ...
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2answers
40 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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0answers
31 views

Largest algebra in a vector space

Let $V$ be a vector space and let $C$ be a subset of $V$ that is closed under a bilinear operator $\langle \cdot, \cdot \rangle: V \times V \rightarrow V$. Let $A \subset V$ be an algebra containing ...
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0answers
54 views

Irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$ [on hold]

We wish to find all the irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$. I came across this problem in my course on advanced algebra. I have little knowledge ...
1
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0answers
14 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 $n-1$, ...
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1answer
44 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 ...
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votes
3answers
37 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 + ...
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votes
1answer
39 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 $0$?
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1answer
22 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 ...
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0answers
19 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 $2^n-1$ ...
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2answers
39 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
52 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
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0answers
29 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
68 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)}{ ...
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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
36 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
44 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
39 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.
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votes
1answer
27 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
34 views

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

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
141 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 ...
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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
21 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 ...
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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
36 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 ...
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0answers
15 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 ...
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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 ...
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0answers
31 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 ...
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1answer
30 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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votes
3answers
82 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 ...
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votes
1answer
35 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 ...
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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
73 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
29 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
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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 ...