All Questions
9
questions
0
votes
1
answer
117
views
Irreducible polynomial with degree 2 over $\mathbb{R}$
Let $f(x) \in \mathbb{R}[x]$ with $\deg f=2$. Show that $f(x)$ is irreducible over $\mathbb{R}$ if and only if $f(x)=(x-a)^2 +b^2$, where $a,b \in \mathbb{R}$ and $b \neq 0$.
I don't know how to do ...
0
votes
2
answers
117
views
Nontrivial subring of $\mathbb{R}$ not containing $1$
Are there examples of nontrivial subrings of $\mathbb{R}$ that do not contain $1$? If not, how can we prove they don't exist? The definition of "ring" here is really "rng"; rings ...
- 1,751
0
votes
1
answer
150
views
Showing a set is an ideal in a ring of real-valued functions
If $F$ is a ring of all real-valued functions defined on $\mathbb{R}$, is $S = \{f ∈ F | f(0) = 1\}$ an ideal?
What I'm thinking is $(f+g)(0) = f(0)+g(0) = 1+1 = 2$
and hence $f + g$ is in $S$? Is ...
- 819
3
votes
1
answer
485
views
Homomorphism from $\mathbb R^2\to \mathbb C$
Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows:
$(a,b)(c,d)=(ac,bd)$
- 4,928
6
votes
4
answers
207
views
Is $\mathbb Q$ a quotient of $\mathbb R[X]$?
Is there some ideal $I \subseteq \mathbb R[X]$ such that $\mathbb R[X]/I \cong \mathbb Q$?
$I$ is clearly not a principal ideal.
- 10.5k
0
votes
3
answers
121
views
Prove $\mathbb{R}$ does not contain a subring isomorphic to $\mathbb{C}$
I'm trying to prove that the quaternions ring $\mathbb{H}$ is not a $\mathbb{C}$-algebra, so I assume $\mathbb{H}$ actually is a complex algebra and that implies that there exists an injective ring ...
- 1,261
1
vote
2
answers
100
views
Is the element $(0,0,0)\in\mathbb{R}^3$ a divisor of zero?
(I'm assuming that $\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}$)
In my assignment, I'm told to prove that exactly one of the following can be true for an element $(x,y,z)\in\mathbb{R}^3$
...
- 2,429
2
votes
0
answers
133
views
Extended real numbers as algebraic structure
I need to work with real numbers, but extended to have an additional element. This element, I denote by $\odot$ and my set is: $\mathbb{R}_{\odot}=\mathbb{R}\cup\{\odot\}$. This element should behave ...
- 165
4
votes
1
answer
455
views
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? [duplicate]
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?
Updated: Is there an example to ...
- 420