Show that the set of real numbers of the form a+b √(2), where a and b are rational, is an ordere

Show that the set of real numbers of the form a+b √(2),...

Key Concepts

Field

A field is an algebraic structure consisting of a set equipped with two operations, addition and multiplication, that satisfy certain axioms including closure, associativity, commutativity, distributivity, and the existence of additive and multiplicative identities and inverses. These properties allow for the manipulation of elements in a consistent algebraic manner and form the basis for much of modern algebra and number theory.

Ordered Field

An ordered field is a field that is also endowed with a total ordering (typically denoted by <) which is compatible with the field operations. This means that the order respects addition and multiplication, allowing one to compare the size of elements in a way that is consistent with the arithmetic structure of the field. The compatibility conditions ensure that if one number is greater than another, adding the same element to both preserves the inequality and multiplying by a positive element does not reverse it.

Subfield of the Real Numbers

A subfield of the real numbers is any subset of the reals which itself forms a field under the usual operations of addition and multiplication. This concept is important because it allows one to study smaller, more manageable systems that still retain the rich algebraic properties of the real number system. Such subfields often provide insight into both the general structure of fields and specific applications in number theory and algebra.

Closure Under Operations

Closure under operations in an algebraic structure means that performing the defined operations (such as addition and multiplication) on any two elements of the set always yields another element that is also in the set. For a set to be considered a field, it must be closed under these operations as well as under the formation of inverses for all nonzero elements. This closure is crucial in verifying that the algebraic structure is self-contained and that the operations are well-defined over the set.

Quadratic Field Extension

A quadratic field extension refers to a field that is constructed by adjoining a square root of a number (typically not a perfect square in the base field) to an existing field. In such an extension, every element can typically be expressed as a linear combination of 1 and the square root, with coefficients in the base field. This concept is fundamental in understanding how more complex fields can be built from simpler ones and plays a significant role in algebraic number theory and the study of algebraic extensions.

Transcript

00:01 Okay, given the set, i mean, let's say our set, our set is called f prime, defined as a plus b times square root of q, where a, b are both rational numbers.

00:24 Okay, first we know f prime is a subset of the real number, and then this subset will inherit the ordered property from the real line, because we know the real line is totally ordered, so as a subset, it will inherit this totally ordered property so we don't need to show f prime is an ordered set, we only need to show f is f prime, it's a field.

01:03 Okay, to show it's a field, we want to show the first f prime is closed, i mean it is closed with respect to two operations, the first one is summation, the second one is the multiplication.

01:23 Okay, field, second, zero, leaving f prime, and if we delete the zero element from the f prime, we know f prime is a group with respect to the multiplication, right, that means for every non -zero element in f prime, it is invertible in f prime.

02:06 So, let's first consider the closeness, it's easy to see for any a1 plus b1 in f prime, and a2 plus b2 times square root of two, for any two elements in this set, if we sum them all together, we'll get a1 plus a2 plus b1 plus b2 times square root of two, which is again an element in our f, because the summation of two rational numbers is again a rational number.

02:47 Okay, so it is closed under summation, that's easy, now we want to show it is closed under multiplication.

02:54 Now, if we multiply them all together, what we get, b1 a2 plus b2 times square root of two, we'll get a1 a2 plus b1 a2 times square root of two plus a1 b2 times square root of two plus b1 b2 times the square of the square root of two, which is equal to two.

03:28 Combining all of those terms, we get, we have a1 a2 plus two times b1 b2 plus a1 b2 plus a2 b1 times square root of two.

03:52 Okay, as a linear combination of rational numbers, or to be more specific, as a rational linear combination with some rational numbers, this guy is again a rational number, so is this guy.

04:18 So that means, if we multiply those two different elements altogether, we again get an element in f prime.

04:28 That means f prime is closed under the multiplication.

04:32 So for the first condition, it has been checked...

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