Let αbe a zero of x^3+x^2+1 in some extension field of Z2. Find the multiplicative inverse of α+

Let αbe a zero of x^3+x^2+1 in some extension field of Z2....

Key Concepts

Field Extensions

A field extension is a larger field that contains a smaller field as a subfield, created by adjoining new elements that satisfy certain polynomial equations. This concept allows one to construct and work within fields that solve equations which are unsolvable in the original smaller field, and it is fundamental in algebra, number theory, and many areas of mathematics.

Irreducible Polynomials

An irreducible polynomial over a given field is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials over that field. Irreducible polynomials are used to define field extensions, as they serve as the minimal polynomials of elements that are adjoined to construct the extension, ensuring unique representation of the elements.

Polynomial Rings and Quotient Rings

A polynomial ring is a set of polynomials with coefficients in a given field, equipped with polynomial addition and multiplication. When a polynomial ideal, generated by an irreducible polynomial, is used to form a quotient ring, the result is a field. This technique is key to constructing finite fields and other extension fields, where elements are represented by polynomial residues modulo the irreducible polynomial.

Multiplicative Inverses in Fields

In any field, every nonzero element has a multiplicative inverse, meaning there exists another element that, when multiplied with the given element, yields the multiplicative identity. The concept is central to solving equations and ensuring the division is always possible in field arithmetic.

The Extended Euclidean Algorithm for Polynomials

The Extended Euclidean Algorithm is a method used to compute the greatest common divisor (gcd) of two polynomials, as well as coefficients (polynomial inverses) that express this gcd as a linear combination of the given polynomials. When applied in the context of a polynomial ring modulo an irreducible polynomial, it can be used to find the multiplicative inverse of a polynomial element, which is crucial for performing division in finite fields.

Transcript

00:02 So what we're going to need to do here is we've got a matrix a and we want to find a squared inverse and a cubed inverse.

00:10 So to start this out, i'm going to say we need to find a inverse.

00:14 The way that we're going to do a inverse for this one, i would recommend doing row reductions.

00:26 So if we write out our matrix and append onto it the identity matrix, we can see that we can subtract two times the bottom row from the middle row.

00:42 Divide the top row by two.

00:45 So if we divide first the top row by two, the only things that are going to change is the two will go to a one, and the one will go to a half.

00:55 And if we take two times this bottom row, we'll get a two here, and if we subtract it from, and a two here.

01:01 And if we subtract that from the middle row, we'll get a zero here and a negative two here.

01:11 So after those row reductions, the right -hand side looks like the identity matrix.

01:15 And so the left -hand side, is just our inverse.

01:18 So a inverse is one -half zero zero zero one negative two zero one.

01:26 From here we need to make note of this important idea that a squared inverse is equal to a inverse squared because when we're working with exponents we multiply them together.

01:40 So both of these equations, both sides of this equation basically says a to the negative two power...

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