Rings and Fields

The complex numbers (ℂ) form a commutative ring with unity under the operations of addition (+) and multiplication (×), where unity is represented by the number 1.

In the field (ℤ/5ℤ, +, ×), the multiplicative inverse of an element a is the element b such that a × b = 1 (mod 5). For 3, we have 3 × 2 = 6 = 1 (mod 5), so the multiplicative inverse of 3 is 2.

A field is a commutative ring with unity in which every nonzero element has a multiplicative inverse. The rational numbers (ℚ) satisfy these properties, making them a field.

The characteristic of a field is the smallest positive integer n such that n × 1 = 0. In the field (𝔽_5, +, ×), we have 5 × 1 = 0, so the characteristic is 5.

An ideal of a ring is a non-empty subset that is closed under addition, multiplication by elements of the ring, and contains the additive identity. (2ℤ, +, ×) satisfies these properties, making it an ideal of (ℤ, +, ×).

The maximal ideal of a ring is the unique largest proper ideal. In the ring (ℤ, +, ×), the only proper ideal is (0ℤ, +, ×), which is also the largest, making it the maximal ideal.

A principal ideal domain is a commutative ring in which every ideal is generated by a single element. The integers (ℤ) form a principal ideal domain because every ideal in (ℤ, +, ×) can be generated by a single integer.

The unity element of a ring is the element that acts as the multiplicative identity. In the ring (ℤ/6ℤ, +, ×), the unity element is 1, since 1 × a = a × 1 = a for any element a in the ring.

A field extension is a field that contains another field as a subfield. The field (ℚ(√2), +, ×) is a field extension of (ℚ, +, ×) because it contains (ℚ, +, ×) as a subfield.

The degree of a field extension is the dimension of the extension field as a vector space over the base field. The degree of (ℚ(√2), +, ×) over (ℚ, +, ×) is 2 because (ℚ(√2), +, ×) is a two-dimensional vector space over (ℚ, +, ×).

A Galois field is a finite field whose order is a prime power. The field (ℤ/5ℤ, +, ×) is a Galois field because its order is 5, which is a prime power.

The order of a Galois field is the number of elements in the field. The order of (ℤ/7ℤ, +, ×) is 7 because it contains 7 distinct elements.

A polynomial ring over a field is the set of all polynomials with coefficients in that field. The polynomial ring (ℤ/2ℤ[x], +, ×) consists of all polynomials with coefficients in the field (ℤ/2ℤ, +, ×).

The degree of a polynomial is the highest exponent of the variable in the polynomial. The degree of x^3 + x + 1 in (ℤ/2ℤ[x], +, ×) is 3.

A maximal ideal of a polynomial ring is a proper ideal that is not contained in any other proper ideal. The ideal (xℤ/2ℤ[x], +, ×) is a maximal ideal of (ℤ/2ℤ[x], +, ×) because it is a proper ideal and it is not contained in any other proper ideal.