Intereting Posts

Links between difference and differential equations?
Prove that $\log _5 7 < \sqrt 2.$
Space of Complex Measures is Banach (proof?)
Calculate half life of esters
A simple permutation question – discrete math
If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.
Determinant of the character table of a finite group $G$
A characterisation of quadratic extensions contained in cyclic extensions of degree 4
Mental card game
Books that every student “needs” to go through
What's the point of orthogonal diagonalisation?
Possibly not an acceptable proof for uncountablity of countable product of countable sets
Measure on the set of rationals
What is the residue of this essential singularity?
$q$-series identity

Prove that the set of symbols $\{a+bi \mid a, b \in \mathbb{F}_3\}$ forms a field with nine elements, if the laws of composition are made to mimic addition and multiplication of complex numbers. Will the same method work for $\mathbb{F}_5$? For $\mathbb{F}_7$?

I was able to prove that $\{a+bi \mid a, b \in \mathbb{F}_3\}$ is a field and that the method does not work for $\mathbb{F}_5$. But could someone explain to me why it works in $\mathbb{F}_7$ ?

Thank you

- Is there a purely algebraic proof of the Fundamental Theorem of Algebra?
- Irreducibility of $x^{n}+x+1$
- Irreducible polynomial which is reducible modulo every prime
- If a field extension is separable, then $=_s$?
- If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$
- Are distinct prime ideals in a ring always coprime? If not, then when are they?

- Irreducibility of $x^{n}+x+1$
- Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.
- Show that the set $\mathbb{Q}(\sqrt{p})=\{a+b\sqrt{p}; a,b,p\in\mathbb{Q},\sqrt{p}\notin \mathbb{Q}\}$ is a field
- Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?
- Splitting field of a separable polynomial is separable
- Why algebraic closures?
- If $b$ is algebraic over a finite extension $K$ of $F$ then $\mid $
- Determine the minimal polynomial of $\sqrt 3+\sqrt 5$
- Abstract algebra book recommendations for beginners.
- Intermediate fields of a finite field extension that is not separable

Everything is completely straightforward over $\mathbb{F}_p$, for any prime $p$, with the possible exception of showing that every nonzero element has a multiplicative inverse.

If you remember how to express reciprocals $\frac1{a + bi}$ by “rationalizing the denominator” (multiply numerator and denominator by the conjugate $a – bi$), then the idea is that this should work here too, as long as $a^2 + b^2$ is guaranteed to be non-zero (as long as one of $a, b$ is nonzero modulo $7$). So you have to show that $a^2 +b^2 = 0$ has no solutions modulo $7$ besides $a = 0 = b$ (modulo $7$, of course). Can you show this is equivalent to $-1$ being a nonsquare modulo $7$?

- Eigenvalue of an Euler product type operator?
- Bases having countable subfamilies which are bases in second countable space
- Limit in Definition of Riemann Integral is one-sided?
- find all self-complementary graphs on five vertices
- Are Mersenne prime exponents always odd?
- Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area
- $G/H$ is a finite group so $G\cong\mathbb Z$
- Prob. 26, Chap. 5 in Baby Rudin: If $\left| f^\prime(x) \right| \leq A \left| f(x) \right|$ on $$, then $f = 0$
- Why is that the extended real line $\mathbb{\overline R}$ do not enjoy widespread use as $\mathbb{R}$?
- Definite integrals with interesting results
- Hockey-Stick Theorem for Multinomial Coefficients
- Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$
- (Certain) colimit and product in category of topological spaces
- Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$.
- Is there a discontinuous function on the plane having partial derivatives of all orders?