Is u Algebraic Over K if u^2 is Algebraic Over F?

jeffreydk · Apr 2, 2009

I'm currently trying to prove that (for a field extension [itex]K[/itex] of the field [itex]F[/itex]) if [itex]u\in K[/itex] and [itex]u^2[/itex] is algebraic over [itex]F[/itex] then [itex]u[/itex] is algebraic over [itex]K[/itex].

I thought of trying to prove it as contrapositive but that got me nowhere--it seems so simple but I don't know what to use for this. Any help with this would be greatly appreciated.

eok20 · Apr 2, 2009

Well clearly u is algebraic over K (it is an elemnt of K) so I'm guessing you mean to say that u is algebraic over F. Well if u^2 is algebraic over F then let f be a polynomial in F[x] such that f(u^2)=0. I don't want to give it away but if you think about the polynomial f(x^2)...

jeffreydk · Apr 2, 2009

Oh wow yea, ok it's pretty clear. I think I was complicating things. Thanks.

Is u Algebraic Over K if u^2 is Algebraic Over F?

Discussion

Undergrad Spinor calculus

Undergrad Matrix representation of rank-2 spinors

Undergrad Looking for a paper about spinors

Undergrad Why are these cycles disjoint and of equal length?

On the Moore–Penrose inverse from a banal linear algebra viewpoint

Undergrad Correct constructed example of Abelian Monoid?

Undergrad Correct state row entries in the following Sequential machine?

Insights Why Vector Spaces Explain The World: A Historical Perspective

Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals

Insights Revisiting the Velocity-Time Function

Insights Remote Operated Gate Control System

Insights AI Enriched Problem Solving

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Is u Algebraic Over K if u^2 is Algebraic Over F?

Discussion

Similar threads

Is t Algebraic Over F(S) in Field Extension E/F?

Proof: $K[a]$ is a Field & if $K(a)=K[a]$, then $a$ is Algebraic Over $K$

Graduate Show K⊂R⊂F Fields: Algebraic Over K => R is a Field

Graduate Is every element in F algebraic over Q?

Is u Algebraic over Extension Field E?

Graduate How Do You Calculate the Dimension of F(u,v) Over F(u)?

Graduate Let E be an algebraic over F, F is perfect. Show that E is perfect

Prove that an element is algebraic over a field extension

Show √ 2 + √ 3 algebraic over Q

Proving Field Properties of R if F is Algebraic over K