(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Can you please check my answers and help me develop explanation for the ones I can't explain.

a) every field has nontrivial extensions.

b) every field has nontrivial algebraic extensions.

c) every simple extension is algebraic.

d) every extension is simple.

e) all simple algebraic extensions of a given subfield of C are isomorphic.

f) all simple transcendental extensions of a given subfield of C are isomorphic.

g) every minimal polynomial is monic.

h) monic polynomails are always irreducible.

i) every polynomial is a constant multiple of an irreducible polynomial.

2. The attempt at a solution

a) true I am thinking of C and adding something that's not in it. like a cube rotation or something call it J. a in C. a+J is a*90degree rotation of the cube. a*J is number of flips. and J+J is a 180 rotation. J*J is again a flip.

b) false. a flip wouldn't be a solution to anything in C. so it has to be a transcendental extension.

c) false Q(pi) is simple and transcendetal.

d) false. only true for when the field has characteristic 0.

e) false? need explanation

f) true? need explanation

g) true. definition of minimal polynomial was that it's monic.

h) false. x^2+4x+2

i) false. need explanation

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# Homework Help: Algabraic field extensions (true false questions)

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