If a is transcendental over F, show that every element of F(a) that is not in F is transcendenta

If a is transcendental over F, show that every element of...

Key Concepts

Field Extension

A field extension is a larger structure containing a base field, formed by adjoining new elements to the base field. In this context, F(a) is an extension of F where the elements are built from F and the element a, allowing operations like addition, subtraction, multiplication, and division to be performed. Field extensions are central in studying how additional elements interact with the existing field structure.

Transcendental Element

A transcendental element is one that is not a root of any non-zero polynomial with coefficients from the base field. In other words, it does not satisfy any algebraic equation with the given field's coefficients. This property distinguishes transcendental elements from algebraic elements, which do satisfy such polynomial equations, and is a key concept when analyzing field extensions like F(a).

Rational Function Field

In the context of a field extension, a rational function field consists of all expressions that can be written as ratios of polynomials in the transcendental element, with coefficients in the base field F. The field F(a) is precisely this kind of structure, and understanding it helps in grasping why elements constructed via non-trivial combinations of a remain transcendental.

Algebraic vs Transcendental Elements

The contrast between algebraic and transcendental elements is fundamental in field theory. An algebraic element satisfies some non-zero polynomial with coefficients in the base field, whereas a transcendental element does not. The problem uses this distinction to show that if a is transcendental, then any element of F(a) that isn’t in F must also be free from any algebraic relation over F, underscoring its transcendental nature.

Transcript

00:03 Section 1 .4, problem number 118, they give us a function here that says f of x is zero when x is rational, f of x is one, when x is irrational, prove this is not continuous for any real number.

00:20 Okay.

00:21 So let's look at our definition of continuity.

00:23 When you start out, definition of continuity is you can, you draw a function and you never lift your pencil.

00:28 Okay, you get a little bit more complicated in calculus where you start to define this with algebraic conditions.

00:34 So for every epsilon greater than zero, real number, that means the y, the y neighborhood.

00:41 For every epsilon, there is a delta, an x neighborhood such that if x minus a is less than some small delta, then f of x minus f of a is less than epsilon.

00:52 So no matter how small a y neighborhood i get, i can pick an x neighborhood that keeps the values within those boundaries.

01:00 Okay, so what we're going to do is going to prove this by contradiction.

01:04 Let epsilon equal 1.

01:06 I should be able to make this work for every epsilon, but i'm going to pick epsilon equal to 1.

01:12 Now, it means that there is a delta such that the absolute value of x minus a is less than delta, which means x minus a is less than delta or minus delta, which means x is between a plus delta and a minus delta.

01:39 So x is a number between a minus delta and a plus delta.

01:44 So let's take the first case when a is rational.

01:50 Okay, so let's go and let's just say that i have a is a rational number.

02:04 Okay, so if a is rational, i know that f of a is going to be zero.

02:11 So this means that f of a is equal to zero.

02:16 So if i look at picking a point x, i need to pick a point this between a minus delta and a plus delta.

02:35 So what i can do by the, so i know that by the density of my irrational numbers, pick x to be an irrational number.

03:20 So what that means is i've got any interval, there are infinite number of rational numbers.

03:24 And irrational numbers in between those two numbers.

03:28 So if i do this, then i've got epsilon that i picked was epsilon equal to 1.

03:36 So f of x minus f of a.

03:40 I need to prove that that is less than 1...

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